Exercise - Build Your Future

1. Question

Find the quotient \(q\) and remainder \(r\) when 67 is divided by 8.

\(q = 7, , r = 5\)

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\(q = 9, , r = -5\)

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\(q = 8, , r = 3\)

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\(q = 6, , r = 11\)

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2. Question

Perform the Euclidean division of \(-45\) by 7. What is the result?

\(q = -6, , r = -3\)

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\(q = -6, , r = 3\)

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\(q = -7, , r = -4\)

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\(q = -7, , r = 2\)

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3. Question

What does the Division Algorithm for integers state?

Every integer \(a\) is divisible by 1 and itself only.

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For every integer \(a\) and positive \(b\), there exists \(r\) such that \(a = r \bmod b\).

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For every integer \(a\) and every positive integer \(b\), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\) with \(0 \leq r < b\).

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There exists only one pair \((q, r)\) such that \(a = qb – r\).

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4. Question

Why is the remainder in an Euclidean division always unique?

Because two remainders will always be zero.

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The quotient compensates for any difference in remainders.

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If \(a = bq + r = bq’ + r’\), then \(b(q – q’) = r’ – r\); since \(|r’ – r| < b\), the only possible multiple of \(b\) in this range is 0, so \(r = r'\) and \(q = q'\).

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Because \(q\) and \(r\) must always be integers.

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5. Question

Without long division, what is the remainder when 213 is divided by 9?

\(7\)

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\(6\)

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\(3\)

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\(2\)

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6. Question

What is the definition of “d is a proper divisor of n”?

\(d \mid n\) with \(d \leq 0\)

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\(d \mid n\) and \(d = n\)

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\(d \mid n\) and \(d = 1\)

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\(d \mid n\) with \(d > 0\) and \(d \ne n\)

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7. Question

What are the proper divisors of 72?

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36

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1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

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2, 4, 6, 8, 12, 18, 36, 72

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1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36

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8. Question

True or False: If \(d \mid a\) and \(d \mid b\), then \(d \mid (a – b)\).

True, because \(a = dk\), \(b = dl \Rightarrow a – b = d(k – l)\)

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False, only holds if \(a > b\)

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False, only holds if \(a = b\)

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False, only for positive \(d\)

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9. Question

Give an integer \(n\) such that \(6 \mid n\) but \(4 \nmid n\).

\(n = 12\)

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\(n = 24\)

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\(n = 36\)

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\(n = 18\)

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10. Question

Prove transitivity of divisibility in one sentence.
If \(a \mid b\) and \(b \mid c\), then:

\(c \mid a\)

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\(a \mid c\) because \(b = ak\), \(c = b\ell = a(k\ell)\)

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\(a = c\)

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\(b = a + c\)

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11. Question

Why is 1 not considered a prime number?

Because it is an odd number.

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Because it has no proper divisors.

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Declaring 1 prime would break the uniqueness in the Fundamental Theorem of Arithmetic.

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Because it is divisible by every number.

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12. Question

Determine whether 161 is a prime number.

161 is not prime because \(161 = 7 \times 23\).

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161 is prime because no prime number divides it.

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161 is prime because it is odd.

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161 is not prime because it is divisible by 9.

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13. Question

Why does every composite number have a prime divisor \(\leq \sqrt{n}\)?

All composite numbers are divisible by 2.

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Because the product of any two numbers exceeds \(n\).

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Because primes are greater than \(n\).

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If \(n = ab\) with \(1 < a \leq b < n\), then \(a \leq \sqrt{n}\).

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14. Question

What is the prime factorisation of 495?

\(5 \cdot 11 \cdot 13\)

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\(3 \cdot 3 \cdot 5 \cdot 11\)

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\(495 = 5 \cdot 3^2 \cdot 11\)

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\(495 = 3 \cdot 7 \cdot 5 \cdot 1\)

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15. Question

Why is 2 the only even prime number?

Because it’s not divisible by any odd number.

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Because all odd numbers are composite.

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Because 2 is not divisible by 3.

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Any even number \(> 2\) has 2 as a proper divisor and is therefore composite.

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16. Question

Compute \(\gcd(84, 98)\) by prime factors.

\(14\) because \(84 = 2^2 \cdot 3 \cdot 7\) and \(98 = 2 \cdot 7^2\), so minimum powers give \(2^1 \cdot 7^1 = 14\).

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\(7\) because it’s the only common prime.

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\(2\) because it appears in both factorizations.

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\(28\) because it divides both.

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17. Question

Why is \(\gcd(a, 0) = |a|\)?

Every divisor of \(a\) divides 0, and the greatest such positive divisor is \(|a|\).

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Because 0 is divisible by all numbers.

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Because \(a + 0 = a\).

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Because \(a \mid 0\).

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18. Question

Show that if \(a \mid b\), then \(\gcd(a, b) = |a|\).

\(a \mid b\) implies \(b = ak\) so \(a\) is the only common factor.

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All common divisors divide \(b\).

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\(b \mid a\) too, so \(\gcd\) is \(b\).

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All common divisors of \(a\) and \(b\) divide \(a\); the greatest is \(|a|\).

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19. Question

Find \(\gcd(2^6 \cdot 3^4, , 2^3 \cdot 3^5)\).

\(2^6 \cdot 3^5 = 5832\)

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\(2^3 \cdot 3^4 = 648\)

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\(2^9 \cdot 3^9\)

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\(2^4 \cdot 3^4 = 1296\)

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20. Question

What is the GCD of any integer \(n\) and \(n + 1\)?

\(n\)

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\(n + 1\)

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\(0\)

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\(1\) because their difference is 1

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21. Question

Execute the Euclidean Algorithm to find \(\gcd(412, 260)\). What is the final result?

\(2\)

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\(6\)

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\(4\)

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\(8\)

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22. Question

How many remainder steps (iterations) were used in the Euclidean Algorithm for \((412, 260)\)?

4

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6

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5

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3

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23. Question

What are the Bézout coefficients for 412 and 260 such that \(4 = 412x + 260y\)?

\(x = 3, , y = -2\)

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\(x = 9, , y = -15\)

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\(x = 1, , y = 2\)

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\(x = -9, , y = 15\)

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24. Question

True or false: \(\gcd(a, b) = \gcd(b, a – b)\).

False

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True – this is the Euclidean step

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False, only if \(a > b\)

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True, but only when \(a = b + 1\)

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25. Question

Use Euclid to compute \(\gcd(987, 610)\) (Fibonacci pair). What is the result?

\(610\)

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\(13\)

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\(1\)

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\(0\)

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26. Question

Solve: Find integers \(x\) and \(y\) such that \(1 = 89x + 55y\).

\(x = 21, , y = -34\)

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\(x = -21, , y = 34\)

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\(x = 13, , y = -8\)

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\(x = 5, , y = -8\)

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27. Question

What is the value of \(\gcd(89, 55)\)?

5

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1

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11

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2

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28. Question

What is the least common multiple (LCM) of 15 and 20?

120

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30

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60

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40

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29. Question

What is the relationship between GCD and LCM of two numbers \(a\) and \(b\)?

\(\text{lcm}(a, b) \cdot \gcd(a, b) = ab\)

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\(\text{lcm}(a, b) + \gcd(a, b) = ab\)

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\(\text{lcm}(a, b) = ab\)

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\(\gcd(a, b) = \frac{a}{b}\)

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30. Question

Find the LCM of 6, 8, and 9.

144

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56

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108

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72

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31. Question

What is the least positive integer divisible by both 12 and 18?

108

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24

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36

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72

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32. Question

If \(\text{lcm}(a, b) = 60\) and \(\gcd(a, b) = 6\), what is \(ab\)?

360

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360 (\(ab = \text{lcm}(a, b) \cdot \gcd(a, b)\))

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66

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180

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33. Question

If two numbers have GCD 1, what are they called?

Multiples

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Identical

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Composite

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Coprime

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34. Question

What is the LCM of 14 and 35?

70

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210

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105

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245

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35. Question

If \(a \mid b\), what is \(\text{lcm}(a, b)\)?

\(b\)

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\(a\)

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\(ab\)

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\(\gcd(a, b)\)

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36. Question

Let \(a = 24\) and \(b = 36\). What is \(\gcd(a, b)\)?

6

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18

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12

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24

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37. Question

What is the LCM of 24 and 36?

72

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48

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60

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96

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38. Question

If \(a = 60\) and \(b = 84\), compute \(\gcd(a, b)\).

24

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6

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18

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12

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39. Question

For \(a = 60\) and \(b = 84\), what is \(\text{lcm}(a, b)\)?

420

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360

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120

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180

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40. Question

What is the LCM of any number \(a\) and 1?

0

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1

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\(a^2\)

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\(a\)

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41. Question

What is \(\gcd(0, 0)\)?

0

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1

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Undefined

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\(\infty\)

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42. Question

Which of the following statements about \(\gcd(a, b)\) is always true?

\(\gcd(a, b) = a + b\)

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\(\gcd(a, b) = ab\)

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\(\gcd(a, b) \leq \min(a, b)\)

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\(\gcd(a, b) \geq \max(a, b)\)

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43. Question

What is the smallest positive integer that can be expressed as a linear combination of 30 and 18?

12

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6

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3

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\(\gcd(30, 18) = 6\)

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44. Question

If \(\gcd(a, b) = d\), then which of the following is true?

\(a/d \mid b/d\)

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\(a/d \text{ and } b/d \text{ are coprime}\)

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\(a + b = d\)

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\(a/d = b/d\)

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45. Question

What is the GCD of 13 and 27?

1

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13

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3

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0

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46. Question

Which of the following is a correct use of Bézout’s identity?

\(\gcd(a, b) = a \cdot b\)

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There exist integers \(x\) and \(y\) such that \(\gcd(a, b) = ax + by\)

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\(a = bx + y\)

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\(b = ax + y\)

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47. Question

What is the Bézout form for \(\gcd(99, 78)\)?

\(99 = 1 \cdot 78 + 21\)

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\(99 – 78 = 21\)

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\(99(-11) + 78(14) = 3\)

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\(99 + 78 = 177\)

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48. Question

Which condition guarantees that two integers are coprime?

Their difference is even.

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Their product is a prime number.

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Their GCD is 1.

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They are both odd.

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49. Question

What is the least common multiple of two coprime numbers \(m\) and \(n\)?

1

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\(m + n\)

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\(mn\)

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\(m – n\)

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50. Question

What are the Bézout coefficients \((x, y)\) such that \(1 = 17x + 31y\)?

\(x = 11, , y = -6\)

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\(x = -11, , y = 6\)

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\(x = 6, , y = -11\)

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\(x = 5, , y = -2\)

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51. Question

Railway problem: tracks 720 m and 1080 m long use identical rail pieces. Find the longest possible piece length.

\( \text{gcd}(720, 1080) = 120,\text{m} \)

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\( \text{gcd}(720, 1080) = 180,\text{m} \)

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\( \text{gcd}(720, 1080) = 360,\text{m} \)

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\( \text{gcd}(720, 1080) = 540,\text{m} \)

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52. Question

Two gears have 42 and 66 teeth. How many turns before they align to their starting position?

7 turns of 42-tooth gear, 6 turns of 66-tooth gear

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14 turns of 42-tooth gear, 9 turns of 66-tooth gear

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11 turns of 42-tooth gear, 7 turns of 66-tooth gear

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6 turns of 42-tooth gear, 11 turns of 66-tooth gear

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53. Question

Solve \( 24x + 33y = 6 \) in integers.

\( x = 3 + 11t,\ y = 2 – 8t,\ t \in \mathbb{Z} \)

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\( x = 2 + 11t,\ y = -1 – 8t,\ t \in \mathbb{Z} \)

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\( x = -2 + 11t,\ y = 3 – 8t,\ t \in \mathbb{Z} \)

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\( x = 2 + 8t,\ y = -1 – 11t,\ t \in \mathbb{Z} \)

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54. Question

Find the smallest positive solution to \( 19x \equiv 1 \pmod{81} \).

\( x = 19 \)

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\( x = 17 \)

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\( x = 81 \)

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\( x = 64 \)

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55. Question

A till contains only €2 and €5 coins. How many ways can you make exactly €54?

6 solutions

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5 solutions

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4 solutions

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7 solutions

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56. Question

Show that the equation \( ax + by = c \) has exactly \( \frac{|c|}{\gcd(a,b)} \) solutions with \( 0 \leq x < \frac{|b|}{d} \), where \( d = \gcd(a,b) \) and \( d \mid c \).

There are \( \gcd(a, b) \) total solutions.

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There are \( \frac{|c|}{\gcd(a, b)} \) total solutions with \( 0 \leq x < \frac{|b|}{\gcd(a, b)} \).

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There are \( \frac{|a|}{\gcd(a, b)} \) total solutions.

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There are \( \frac{|c|}{|b|} \) solutions only when \( a \mid c \).

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57. Question

Use exercise 56 to count the solutions in non-negative integers of \( 7x + 5y = 70 \).

5

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3

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2

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4

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58. Question

Find all integer triples \( (x, y, z) \) satisfying:
\( x + y = 31,\quad \gcd(x,z) = 1,\quad z \mid y. \)

62

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34

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33

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31

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59. Question

(Olympiad) Prove that the equation \( \frac{1}{x} + \frac{1}{y} = \frac{1}{n} \) with \( x \leq y \) has exactly \( d(n^2) \) distinct positive solutions, where \( d(m) \) is the divisor-count function.

\( d(n^2) \)

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\( \frac{1}{2} d(n^2) \)

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\( d(n) \)

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\( 2d(n) \)

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60. Question

List all solutions for \( n = 6 \) in exercise 59 (i.e., \( \frac{1}{x} + \frac{1}{y} = \frac{1}{n} \) with \( x \leq y \)).

4

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5

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6

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9

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61. Question

Prove that if \( \gcd(a,b) = 1 \) and \( k \mid (a + b) \) with \( k \mid ab \), then \( k = 1 \) or \( k = ab \).

\( k = a + b \) always

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\( k = a – b \) if \( k \mid a+b \)

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\( k = 1 \) or \( k = ab \)

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\( k = \gcd(a, ab) \)

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62. Question

Let \( p, q \) be distinct primes. Show \( \gcd(p^m – 1, q^n – 1) = 1 \) unless both \( m \) and \( n \) are even.

When both \( p \) and \( q \) are odd

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When both \( m \) and \( n \) are even

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When \( p = q \)

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When \( m = n \)

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63. Question

Prove that the product of four consecutive integers is always divisible by 24.

Two numbers are divisible by 3

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One number is divisible by 4

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Two are even, one divisible by 4, one divisible by 3

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All numbers are prime

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64. Question

Show that \( a^2 – b^2 \) and \( a + b \) are always coprime or differ by at most a factor of 2.

Their gcd is always 1

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Their gcd divides 2

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Their gcd is \( a + b \)

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Their gcd divides \( a^2 + b^2 \)

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65. Question

Prove that a perfect square has an odd number of positive divisors.

All numbers have even number of divisors

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Perfect squares have more than 10 divisors

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Only odd numbers are divisors of squares

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The square root occurs only once, so the divisor count is odd

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66. Question

How many divisors does 900 have?

18

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36

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12

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27

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67. Question

Compute \( \phi(96) \) (Euler’s totient function).

192

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32

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36

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48

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68. Question

Perform the division of \( f(x) = 2x^3 – 3x^2 + 4x – 6 \) by \( g(x) = x – 2 \) over \( \mathbb{Q} \).

Quotient: \( 2x^2 + x + 4 \), Remainder: \( 0 \)

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Quotient: \( 2x^2 + x + 6 \), Remainder: \( 6 \)

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Quotient: \( x^2 + 2x + 6 \), Remainder: \( 0 \)

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Quotient: \( 2x^2 + x – 6 \), Remainder: \( 12 \)

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69. Question

Apply the polynomial Euclidean Algorithm to \( f(x) = x^2 – 1 \) and \( g(x) = x – 1 \) over \( \mathbb{Q} \).

\( x – 1 \)

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\( x + 1 \)

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\( x^2 – 1 \)

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\( 1 \)

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70. Question

Show that \( x^3 – 1 \) and \( x^4 + 1 \) are coprime over \( \mathbb{Q}[x] \).

They share \( x – 1 \) as a factor

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Both are divisible by \( x + 1 \)

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They have no common root

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They are both irreducible

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71. Question

Compute the Bézout identity for \( x^2 – 2 \) and \( 2x + 2 \) in \( \mathbb{Q}[x] \).

\( A(x)(x^2 – 2) + B(x)(2x + 2) = 1 \)

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\( A(x)(x^2 – 2) + B(x)(2x + 2) = 0 \)

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\( (x^2 – 2)(2x + 2) = 1 \)

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\( A(x)(x^2 – 2) = B(x)(2x + 2) \)

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72. Question

In \( \mathbb{Z}[i] \) (Gaussian integers), what are the units?

\( {1, -1} \)

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\( {i, -i} \)

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\( {1, -1, i, -i} \)

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\( {0, 1, -1} \)

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73. Question

Show that \( 3 + 4i \) and \( 4 – i \) are associates in \( \mathbb{Z}[i] \).

They are conjugates

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They differ by multiplication with a unit

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They are both primes

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They have the same norm

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74. Question

Prove that every non-zero element in a field is a unit.

Only rational fields allow it

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Every non-zero element has a multiplicative inverse

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Units are defined as primes

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Only elements of degree 1 are units

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75. Question

Are 6 and –6 associates in \( \mathbb{Z} \)?

They are both even

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They have the same prime factorization

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They are both divisible by 3

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They differ by multiplication with a unit

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76. Question

Three traffic lights flash every 45, 60, and 72 seconds. When will they next flash together?

180 seconds

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720 seconds

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360 seconds

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90 seconds

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77. Question

Show that
\( \frac{\text{lcm}(a, b) \cdot \gcd(a, b)}{ab} = 1 \)

It holds only for prime \( a \) and \( b \)

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It follows from comparing maximum and minimum prime exponents

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Because \( \gcd(a, b) = 1 \)

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It holds only if \( a = b \)

Correct Incorrect Correct answer

78. Question

In the binomial row \( \binom{5}{0}, \binom{5}{1}, \ldots, \binom{5}{5} \), the distinct values are \( 1, 5, 10 \). What is their LCM?

5

Correct Incorrect Correct answer

15

Correct Incorrect Correct answer

20

Correct Incorrect Correct answer

10

Correct Incorrect Correct answer

79. Question

A 12-hour and an 18-hour tide coincide today. After how many days will the pattern repeat exactly at midnight?

1.5 days

Correct Incorrect Correct answer

2 days

Correct Incorrect Correct answer

3 days

Correct Incorrect Correct answer

6 days

Correct Incorrect Correct answer

80. Question

Compute \( \text{lcm}(24, 28, 98) \).

3024

Correct Incorrect Correct answer

15552

Correct Incorrect Correct answer

1176

Correct Incorrect Correct answer

672

Correct Incorrect Correct answer

81. Question

Show that 561 is composite even though it satisfies \( a^{560} \equiv 1 \pmod{561} \) for every \( a \) coprime to 561.

561 is a prime number

Correct Incorrect Correct answer

561 is a Carmichael number

Correct Incorrect Correct answer

561 is divisible by 7

Correct Incorrect Correct answer

561 is a power of 3

Correct Incorrect Correct answer

82. Question

Determine the remainder when \( 7^{72} \) is divided by 100.

49

Correct Incorrect Correct answer

72

Correct Incorrect Correct answer

0

Correct Incorrect Correct answer

1

Correct Incorrect Correct answer

83. Question

Show there are infinitely many primes congruent to 2 modulo 3.

Contradiction using Fermat’s method and modular arithmetic

Correct Incorrect Correct answer

Contrapositive proof

Correct Incorrect Correct answer

Prime sieve argument

Correct Incorrect Correct answer

Euclid’s proof with even primes

Correct Incorrect Correct answer

84. Question

Solve in integers: \( 2x^2 – 5y^2 = 1 \).

Pell equation; general solution: \( x + y\sqrt{5} = (9 + 4\sqrt{5})^n \)

Correct Incorrect Correct answer

No integer solution

Correct Incorrect Correct answer

Only solution: \( (1,1) \)

Correct Incorrect Correct answer

Quadratic formula gives all roots

Correct Incorrect Correct answer

85. Question

Prove that the equation \( x^3 + y^3 = 2z^3 \) has no positive integer solutions.

Fermat’s Little Theorem

Correct Incorrect Correct answer

Prime factorization

Correct Incorrect Correct answer

Modular arithmetic

Correct Incorrect Correct answer

Infinite descent argument (Euler’s method)

Correct Incorrect Correct answer

86. Question

Show that \( \binom{n}{k} \) is an integer for all integers \( n, k \) with \( 0 \leq k \leq n \).

Because factorials are integers

Correct Incorrect Correct answer

Because it counts selections (combinatorial interpretation)

Correct Incorrect Correct answer

Because \( n^k \) is divisible by \( k! \)

Correct Incorrect Correct answer

Only true for even \( n \)

Correct Incorrect Correct answer

87. Question

Determine all \( n \) such that \( n \mid 2^n – 2 \).

Only prime \( n \)

Correct Incorrect Correct answer

All \( n \) dividing \( 2^n – 2 \); includes Carmichael numbers

Correct Incorrect Correct answer

Powers of 2 only

Correct Incorrect Correct answer

Only \( n = 1, 2, 3, 6 \)

Correct Incorrect Correct answer

88. Question

Prove that the product of any two consecutive Fibonacci numbers is divisible by their index.

Fibonacci recurrence

Correct Incorrect Correct answer

Binet’s formula

Correct Incorrect Correct answer

Cassini’s identity and induction on \( n \)

Correct Incorrect Correct answer

Euclidean lemma

Correct Incorrect Correct answer

89. Question

Show that \( x^2 \equiv -1 \pmod{p} \) is solvable iff \( p \equiv 1 \pmod{4} \).

When \( p \equiv 1 \pmod{4} \)

Correct Incorrect Correct answer

For all odd primes

Correct Incorrect Correct answer

When \( p \equiv 3 \pmod{4} \)

Correct Incorrect Correct answer

Never

Correct Incorrect Correct answer

90. Question

(China 1991) Solve \( x^2 + y^2 + z^2 = 2xyz \) in positive integers.

\( (2, 2, 2) \)

Correct Incorrect Correct answer

\( (3, 3, 1) \)

Correct Incorrect Correct answer

\( (2, 1, 1) \)

Correct Incorrect Correct answer

\( (1, 1, 1) \)

Correct Incorrect Correct answer

91. Question

Solve in integers: \( x^2 – 5y^2 = 1 \).

\( x + y\sqrt{5} = (9 + 4\sqrt{5})^n,\ n \in \mathbb{Z}_{\geq 0} \)

Correct Incorrect Correct answer

\( x = 5y + 1 \)

Correct Incorrect Correct answer

\( x = 2y – 3 \)

Correct Incorrect Correct answer

\( x = y^2 – 5 \)

Correct Incorrect Correct answer

92. Question

Prove that the equation \( x^3 + y^3 = z^3 \) has no positive integer solutions.

Fermat’s Last Theorem for \( n = 4 \)

Correct Incorrect Correct answer

Modular arithmetic

Correct Incorrect Correct answer

Infinite descent (Euler’s method)

Correct Incorrect Correct answer

Trial and error with small values

Correct Incorrect Correct answer

93. Question

Find all integer solutions of \( x^2 + 4 = 3y^2 \).

Use \( x = 2y – 1 \)

Correct Incorrect Correct answer

\( x + y\sqrt{3} = (2 + \sqrt{3})^n \)

Correct Incorrect Correct answer

Pell-type via \( \mathbb{Z}[\sqrt{3}] \), minimal solution \( (x,y) = (2,2) \)

Correct Incorrect Correct answer

No integer solutions

Correct Incorrect Correct answer

94. Question

Show that \( \frac{n^2 – n}{6} \) is always an integer.

Because \( n(n-1) \) is even

Correct Incorrect Correct answer

It equals \( \binom{n}{2} \), the number of pairs

Correct Incorrect Correct answer

It’s divisible by 2

Correct Incorrect Correct answer

It equals \( \binom{n}{3} \)

Correct Incorrect Correct answer

95. Question

Determine all \( n \) such that \( n \mid 2^n – 2 \).

All divisors of \( 2^n – 2 \); includes Carmichael numbers

Correct Incorrect Correct answer

Only primes

Correct Incorrect Correct answer

Only powers of 2

Correct Incorrect Correct answer

Even numbers only

Correct Incorrect Correct answer

96. Question

Prove that the product of any two consecutive Fibonacci numbers is divisible by their index.

Binet’s formula

Correct Incorrect Correct answer

Lucas numbers

Correct Incorrect Correct answer

\( F_n F_{n+1} = F_{n+1}F_n – F_{n-1}F_{n-1} \) and Cassini’s identity

Correct Incorrect Correct answer

\( F_{n+1}^2 – F_n^2 = F_{n+2}F_n \)

Correct Incorrect Correct answer

97. Question

Show that \( x^2 \equiv -1 \pmod{p} \) is solvable iff \( p \equiv 1 \pmod{4} \).

Möbius function

Correct Incorrect Correct answer

Jacobi symbol

Correct Incorrect Correct answer

Chinese remainder theorem

Correct Incorrect Correct answer

Legendre symbol: \( \left( \frac{-1}{p} \right) = (-1)^{(p-1)/2} \)

Correct Incorrect Correct answer

98. Question

Find all non-negative integer solutions to \( 4x + 9y + 25z = 50 \).

\( {(2,1,1), (4,2,1)} \)

Correct Incorrect Correct answer

\( {(8,2,0), (4,1,1), (0,0,2)} \)

Correct Incorrect Correct answer

\( {(6,3,0), (3,1,2)} \)

Correct Incorrect Correct answer

\( {(10,0,0), (5,2,1)} \)

Correct Incorrect Correct answer

99. Question

(China 1991) Solve \( x^2 + y^2 + z^2 = 2xyz \) in positive integers.

\( (2, 2, 2) \)

Correct Incorrect Correct answer

\( (3, 3, 3) \)

Correct Incorrect Correct answer

\( (1, 1, 1) \)

Correct Incorrect Correct answer

\( (1, 2, 2) \)

Correct Incorrect Correct answer

100. Question

Prove that for \( n > 3 \), \( n! \equiv 0 \pmod{n} \).

Only if \( n \) is odd

Correct Incorrect Correct answer

Because \( n! \) is a multiple of \( n – 1 \)

Correct Incorrect Correct answer

Only when \( n \) is even

Correct Incorrect Correct answer

Factorial always contains \( n \) for composite or prime \( n \)

Correct Incorrect Correct answer

101. Question

Compute gcd(\(x^4 + x^2 + 1, x^3 + 1\)) over \(\mathbb{Z}[x]\).

\(x + 1\)

Correct Incorrect Correct answer

\(x^2 + 1\)

Correct Incorrect Correct answer

\(x^3 + 1\)

Correct Incorrect Correct answer

\(1\)

Correct Incorrect Correct answer

102. Question

Over \(\mathbb{Q}[x]\), show \(x^5 – 1\) factors as \((x – 1)(x^4 + x^3 + x^2 + x + 1)\). Verify the factor is irreducible.

It has rational roots.

Correct Incorrect Correct answer

It is reducible by factoring as two quadratics.

Correct Incorrect Correct answer

It is divisible by \(x + 1\).

Correct Incorrect Correct answer

It passes Eisenstein’s criterion (at prime 2 on reversed polynomial).

Correct Incorrect Correct answer

103. Question

Find the remainder of \(x^{100}\) on division by \(x^2 + 1\).

\(0\)

Correct Incorrect Correct answer

\(1\)

Correct Incorrect Correct answer

\(x\)

Correct Incorrect Correct answer

\(-1\)

Correct Incorrect Correct answer

104. Question

Compute \(\gcd(x^6 – 1, x^4 – 1)\) over \(\mathbb{Q}[x]\).

\(x^2 – 1\)

Correct Incorrect Correct answer

\((x^2 + 1)\)

Correct Incorrect Correct answer

\((x – 1)^2\)

Correct Incorrect Correct answer

\((x + 1)(x^2 + 1)\)

Correct Incorrect Correct answer

105. Question

Use Euclid to show \(\gcd(3x^2 – 2x + 2, x^2 – x + 1) = 1\).

First polynomial is irreducible.

Correct Incorrect Correct answer

Roots are irrational.

Correct Incorrect Correct answer

Division leads to a constant non-zero remainder.

Correct Incorrect Correct answer

They share no terms.

Correct Incorrect Correct answer

106. Question

In \(\mathbb{C}[x]\), find all polynomials dividing both \(x^{10} – 1\) and \(x^6 – 1\).

\(x^2 – 1\) and its factors

Correct Incorrect Correct answer

\(x^5 – 1\)

Correct Incorrect Correct answer

\(x^{30} – 1\)

Correct Incorrect Correct answer

\(x^6 + x^{10}\)

Correct Incorrect Correct answer

107. Question

Prove \(x^n – 1\) and \(x^m – 1\) are coprime over \(\mathbb{Q}[x]\) if and only if \(\gcd(n, m) = 1\).

When both \(n\) and \(m\) are even

Correct Incorrect Correct answer

When \(\gcd(n, m) = 1\)

Correct Incorrect Correct answer

When \(n = m\)

Correct Incorrect Correct answer

When \(n\) divides \(m\)

Correct Incorrect Correct answer

108. Question

Compute \(\gcd(x^5 – x, x^4 – 1)\).

\(x^2 – 1\)

Correct Incorrect Correct answer

\(x + 1\)

Correct Incorrect Correct answer

\(x^4 – x\)

Correct Incorrect Correct answer

\(x – 1\)

Correct Incorrect Correct answer

109. Question

Show that any common divisor of two monic polynomials can be taken monic.

Because all monic polynomials are irreducible.

Correct Incorrect Correct answer

Because their degrees are the same.

Correct Incorrect Correct answer

Multiply by the reciprocal of the leading coefficient (a unit).

Correct Incorrect Correct answer

Because they share a root at 0.

Correct Incorrect Correct answer

110. Question

State and prove the Division Algorithm for polynomials.

Always divides with zero remainder.

Correct Incorrect Correct answer

Subtract multiples of \(g\) to reduce degree; remainder degree is less than \(g\)

Correct Incorrect Correct answer

Quotient always has lower degree than remainder

Correct Incorrect Correct answer

Works only when \(\deg(f) = \deg(g)\)

Correct Incorrect Correct answer

111. Question

Show that if \(p\) is prime, then \(p\) divides the binomial coefficient \(\binom{p}{k}\) for \(0 < k < p\).

Numerator has factor \(p\), denominator does not

Correct Incorrect Correct answer

Both numerator and denominator divisible by \(p\)

Correct Incorrect Correct answer

Because \(k = p/2\)

Correct Incorrect Correct answer

Binomial coefficient is always zero

Correct Incorrect Correct answer

112. Question

Evaluate \(\gcd\left(\binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n-1} \right)\).

\(1\) always

Correct Incorrect Correct answer

\(\binom{n}{n-1}\)

Correct Incorrect Correct answer

\(n\) if \(n\) is prime, otherwise \(1\)

Correct Incorrect Correct answer

\(n-1\) if \(n\) is composite

Correct Incorrect Correct answer

113. Question

Show that \(x^p – x\) over \(\mathbb{F}_p\) factors as a product of all monic linear polynomials.

It’s irreducible over \(\mathbb{F}_p\)

Correct Incorrect Correct answer

Its degree is too high

Correct Incorrect Correct answer

Every element in \(\mathbb{F}_p\) is a root

Correct Incorrect Correct answer

All polynomials split by default in finite fields

Correct Incorrect Correct answer

114. Question

Prove Wilson’s theorem: \((p – 1)! \equiv -1 \pmod{p}\) for prime \(p\).

It doesn’t hold; it’s a special case

Correct Incorrect Correct answer

Elements pair with their inverses except \(1\) and \(p – 1\)

Correct Incorrect Correct answer

Factorials cancel out mod \(p\)

Correct Incorrect Correct answer

Fermat’s Little Theorem implies this

Correct Incorrect Correct answer

115. Question

Deduce that if \((n – 1)! \equiv -1 \pmod{n}\), then \(n\) is prime.

If \((n – 1)! \equiv 1 \pmod{n}\), then \(n\) is prime

Correct Incorrect Correct answer

This only holds for \(n = 2\)

Correct Incorrect Correct answer

Composite \(n\) also satisfy the congruence

Correct Incorrect Correct answer

If \((n – 1)! \equiv -1 \pmod{n}\), then \(n\) is prime

Correct Incorrect Correct answer

116. Question

Compute \(\gcd(2^{100} – 1, 2^{60} – 1)\).

\(2^{40} – 1\)

Correct Incorrect Correct answer

\(2^{20} – 1\)

Correct Incorrect Correct answer

\(2^{\gcd(100,60)} – 1 = 2^{20} – 1 = 1,048,575\)

Correct Incorrect Correct answer

\(2^{100} – 2^{60}\)

Correct Incorrect Correct answer

117. Question

Let \(F_n\) denote the \(n\)th Fibonacci number. Show \(\gcd(F_m, F_n) = F_{\gcd(m,n)}\).

\(\gcd(F_m, F_n) = F_{\gcd(m,n)}\)

Correct Incorrect Correct answer

\(\gcd(F_m, F_n) = \gcd(m,n)\)

Correct Incorrect Correct answer

\(\gcd(F_m, F_n) = F_{m+n}\)

Correct Incorrect Correct answer

\(\gcd(F_m, F_n) = 1\) if \(m \neq n\)

Correct Incorrect Correct answer

118. Question

Prove that \(6 \mid n(n+1)(n+2)\) for all integers \(n\).

Among three consecutive integers, one divisible by 3 and one by 2

Correct Incorrect Correct answer

All three terms are divisible by 6

Correct Incorrect Correct answer

It’s always a multiple of 12

Correct Incorrect Correct answer

It’s always divisible by \(n\)

Correct Incorrect Correct answer

119. Question

Show \(n^5 – n\) is divisible by 30 for all integers \(n\).

It is divisible by \(n^2 – 1\)

Correct Incorrect Correct answer

Always divisible by 10

Correct Incorrect Correct answer

Only holds for even \(n\)

Correct Incorrect Correct answer

Use Fermat/Chinese Remainder: divisible by 2, 3, 5 individually

Correct Incorrect Correct answer

120. Question

Determine \(\gcd(n!, (n + 1)!)\).

!

Correct Incorrect Correct answer

n!

Correct Incorrect Correct answer

1

Correct Incorrect Correct answer

2

Correct Incorrect Correct answer

121. Question

For integers \(a, b\), prove \(\gcd(a + b, a – b)\) equals 1 or 2.

Because sum and difference are always coprime

Correct Incorrect Correct answer

Since \(a\) and \(b\) are even

Correct Incorrect Correct answer

GCD is always 1 when \(a \ne b\)

Correct Incorrect Correct answer

Their gcd divides both sum and difference, hence divides \(2a\) and \(2b\)

Correct Incorrect Correct answer

122. Question

Show that if \(a \equiv b \pmod{n}\), then \(\gcd(a, n) = \gcd(b, n)\).

Because \(a\) and \(b\) are equal

Correct Incorrect Correct answer

\(a – b\) is divisible by \(n\), so common divisors with \(n\) are the same

Correct Incorrect Correct answer

\(b\) is a multiple of \(n\)

Correct Incorrect Correct answer

GCD doesn’t depend on congruence

Correct Incorrect Correct answer

123. Question

Prove \(\text{lcm}(a,b,c) \leq abc\) with equality if and only if \(\gcd\) is pairwise 1.

When all pairs are coprime

Correct Incorrect Correct answer

Always

Correct Incorrect Correct answer

When \(a + b + c\) is even

Correct Incorrect Correct answer

If all numbers are prime

Correct Incorrect Correct answer

124. Question

Show that if \(a \mid bc\) and \(\gcd(a,b) = d\), then \(\frac{a}{d} \mid c\).

Because \(a \mid b\)

Correct Incorrect Correct answer

Write \(a = bd\ell\); cancel since \(\ell\) is coprime to \(b\)

Correct Incorrect Correct answer

Because \(b \mid a\)

Correct Incorrect Correct answer

Because \(d \mid c\)

Correct Incorrect Correct answer

125. Question

If \(\gcd(a, b) = 1\), prove that \(\gcd\left(a^{m} – b^{m}, a^{n} – b^{n}\right) = a^{\gcd(m,n)} – b^{\gcd(m,n)}\).

\(\gcd(a^m + b^m, a^n + b^n) = 1\)

Correct Incorrect Correct answer

\(a^m – b^m\) always divides \(a^n – b^n\)

Correct Incorrect Correct answer

\(\gcd(a^m – b^m, a^n – b^n) = a^{\gcd(m,n)} – b^{\gcd(m,n)}\)

Correct Incorrect Correct answer

GCD is always \(a – b\)

Correct Incorrect Correct answer

126. Question

Prove that if \(a \mid c\) and \(b \mid c\) and \(\gcd(a,b) = 1\), then \(ab \mid c\).

Since \(a\) and \(b\) are coprime, their product divides common multiple

Correct Incorrect Correct answer

Because \(c = a + b\)

Correct Incorrect Correct answer

Because \(a = b\)

Correct Incorrect Correct answer

Only if \(c\) is even

Correct Incorrect Correct answer

127. Question

Show that for all \(n \geq 1\), \(\gcd(n, n+1) = 1\).

Because both are odd

Correct Incorrect Correct answer

Even numbers can’t be coprime

Correct Incorrect Correct answer

They have the same prime factor

Correct Incorrect Correct answer

No number greater than 1 divides both

Correct Incorrect Correct answer

128. Question

Prove that for any integers \(a, b\), \(\gcd(a, b) = \gcd(b, a \mod b)\).

\(a \mod b = 0 \Rightarrow \gcd = a\)

Correct Incorrect Correct answer

\(b = a \mod b\) always

Correct Incorrect Correct answer

\(\gcd(a, b) = \gcd(b, a \mod b)\)

Correct Incorrect Correct answer

\(\gcd(a, b) = a – b\)

Correct Incorrect Correct answer

129. Question

Show that \(a^2 \equiv 1 \pmod{n}\) implies \((a – 1)(a + 1) \equiv 0 \pmod{n}\).

\(a \equiv 0 \pmod{n}\)

Correct Incorrect Correct answer

\((a – 1)(a + 1) \equiv 0 \pmod{n}\)

Correct Incorrect Correct answer

\(a \equiv 2 \pmod{n}\)

Correct Incorrect Correct answer

\(a \equiv -1 \pmod{n}\)

Correct Incorrect Correct answer

130. Question

Prove that if \(a \equiv b \pmod{n}\) and \(d \mid n\), then \(a \equiv b \pmod{d}\).

Only when \(n = d^2\)

Correct Incorrect Correct answer

Because \(\)a \equiv 0 \pmod{n}[/latex_inline

Correct Incorrect Correct answer

Since \(n = kd\), any multiple of \(n\) is a multiple of \(d\)

Correct Incorrect Correct answer

It does not hold unless \(n \mid d\)

Correct Incorrect Correct answer

131. Question

Show that if \(a \equiv b \pmod{m}\) and \(c \equiv d \pmod{m}\), then \(a + c \equiv b + d \pmod{m}\).

Congruence is preserved under addition

Correct Incorrect Correct answer

\(a + c = b + d\) always

Correct Incorrect Correct answer

Addition mod \(m\) only works when \(a = c\)

Correct Incorrect Correct answer

Modulo \(m\) does not apply to sums

Correct Incorrect Correct answer

132. Question

Prove that \((a + b) \mod m = ((a \mod m) + (b \mod m)) \mod m\).

Only true if \(a = b\)

Correct Incorrect Correct answer

Modular addition distributes over operands

Correct Incorrect Correct answer

Valid only for even \(m\)

Correct Incorrect Correct answer

False if \(a + b > m\)

Correct Incorrect Correct answer

133. Question

Show that \(ab \mod m = ((a \mod m)(b \mod m)) \mod m\).

Modular multiplication distributes over operands

Correct Incorrect Correct answer

Only true if \(a\) and \(b\) are coprime to \(m\)

Correct Incorrect Correct answer

It doesn’t; multiplication breaks congruence

Correct Incorrect Correct answer

Only valid for primes

Correct Incorrect Correct answer

134. Question

Let \(a \equiv b \pmod{m}\). Prove \(a^k \equiv b^k \pmod{m}\) for any \(k \in \mathbb{N}\).

Only true if \(a, b\) are equal

Correct Incorrect Correct answer

Not true unless \(k = 2\)

Correct Incorrect Correct answer

Works only for odd \(k\)

Correct Incorrect Correct answer

Exponentiation preserves congruence

Correct Incorrect Correct answer

135. Question

If \(a \equiv b \pmod{m}\) and \(m \mid ab\), is it true that \(m \mid a\) or \(m \mid b\)?

Always

Correct Incorrect Correct answer

Only if \(a \equiv b \pmod{2m}\)

Correct Incorrect Correct answer

No; counterexample with \(m = 6, a = 2, b = 3\)

Correct Incorrect Correct answer

True for prime \(m\) only

Correct Incorrect Correct answer

136. Question

Prove that for \(a, b \in \mathbb{Z}\), \(a \equiv b \pmod{m} \Rightarrow f(a) \equiv f(b) \pmod{m}\) for any polynomial \(f(x) \in \mathbb{Z}[x]\).

Only linear functions preserve congruence

Correct Incorrect Correct answer

It fails for degrees \( > 2\)

Correct Incorrect Correct answer

Only works if \(a \leq b\)

Correct Incorrect Correct answer

Congruence respects substitution into integer-coefficient polynomials

Correct Incorrect Correct answer

137. Question

Prove that if \(a \equiv b \pmod{m}\) and \(f(x)\) is monic, then \(f(a) \equiv f(b) \pmod{m}\).

Guarantees unique roots mod \(m\)

Correct Incorrect Correct answer

No extra condition needed; congruence works for any integer polynomial

Correct Incorrect Correct answer

Only true if \(f\) has constant term zero

Correct Incorrect Correct answer

Only works for prime \(m\)

Correct Incorrect Correct answer

138. Question

Prove that if \(f(x) \in \mathbb{Z}[x]\), then \(f(a) \equiv f(b) \pmod{m}\) whenever \(a \equiv b \pmod{m}\).

Each power and coefficient respects mod arithmetic

Correct Incorrect Correct answer

Only works for linear terms

Correct Incorrect Correct answer

False unless \(a = b\)

Correct Incorrect Correct answer

Works only for constant \(f(x)\)

Correct Incorrect Correct answer

139. Question

Determine whether \(x^2 + 1\) has roots modulo 5.

No, 5 is not a square mod

Correct Incorrect Correct answer

Only \(x \equiv 1\)

Correct Incorrect Correct answer

Yes, \(x \equiv 2, 3 \pmod{5}\)

Correct Incorrect Correct answer

Yes, but only \(x \equiv 0\)

Correct Incorrect Correct answer

140. Question

Show that \(x^2 + 1\) has no solution modulo 3.

Yes, \(x \equiv 1\)

Correct Incorrect Correct answer

Yes, \(x \equiv 2\)

Correct Incorrect Correct answer

Yes, for \(x \equiv -1\)

Correct Incorrect Correct answer

No; \(x^2 \equiv 0, 1 \pmod{3}\) only

Correct Incorrect Correct answer

141. Question

Determine all \(x\) such that \(x^2 \equiv 1 \pmod{7}\).

\(x \equiv 1, 6 \pmod{7}\)

Correct Incorrect Correct answer

\(x \equiv 2, 5 \pmod{7}\)

Correct Incorrect Correct answer

\(x \equiv 3, 4 \pmod{7}\)

Correct Incorrect Correct answer

\(x \equiv 0, 7 \pmod{7}\)

Correct Incorrect Correct answer

142. Question

Find all \(x\) such that \(x^2 \equiv 2 \pmod{5}\).

\(x \equiv 1, 4 \pmod{5}\)

Correct Incorrect Correct answer

No solution; 2 is not a quadratic residue mod 5

Correct Incorrect Correct answer

\(x \equiv 2, 3 \pmod{5}\)

Correct Incorrect Correct answer

\(x \equiv 0, 5 \pmod{5}\)

Correct Incorrect Correct answer

143. Question

Show that if \(x^2 \equiv a \pmod{p}\) has a solution, then \(a^{(p-1)/2} \equiv 1 \pmod{p}\).

Fermat’s Theorem

Correct Incorrect Correct answer

Wilson’s Theorem

Correct Incorrect Correct answer

Euler’s Criterion

Correct Incorrect Correct answer

Legendre Identity

Correct Incorrect Correct answer

144. Question

If \(a^{(p-1)/2} \equiv -1 \pmod{p}\), what can we conclude about \(a\)?

\(a\) is a non-quadratic residue modulo \(p\)

Correct Incorrect Correct answer

\(a\) is divisible by \(p\)

Correct Incorrect Correct answer

\(a\) is a square modulo \(p\)

Correct Incorrect Correct answer

\(a \equiv 0 \pmod{p}\)

Correct Incorrect Correct answer

145. Question

Determine whether \(3\) is a quadratic residue modulo \(7\).

Yes, \(3^2 \equiv 3 \pmod{7}\)

Correct Incorrect Correct answer

No, \(3^{(7-1)/2} = 3^3 = 27 \equiv -1 \pmod{7}\)

Correct Incorrect Correct answer

Yes, \(3 \equiv 2^2 \pmod{7}\)

Correct Incorrect Correct answer

Only when \(p = 3\)

Correct Incorrect Correct answer

146. Question

Prove that if \(a^2 \equiv b^2 \pmod{p}\), then \(a \equiv \pm b \pmod{p}\) for prime \(p\).

\(a \equiv b \pmod{p}\) or \(a \equiv -b \pmod{p}\)

Correct Incorrect Correct answer

\(a \equiv b + 1 \pmod{p}\)

Correct Incorrect Correct answer

\(a \equiv 0 \pmod{p}\)

Correct Incorrect Correct answer

\(b \equiv 1 \pmod{p}\)

Correct Incorrect Correct answer

147. Question

Determine all solutions of \(x^2 \equiv 4 \pmod{11}\).

\(x \equiv 2, 9 \pmod{11}\)

Correct Incorrect Correct answer

\(x \equiv 3, 8 \pmod{11}\)

Correct Incorrect Correct answer

\(x \equiv 4, 7 \pmod{11}\)

Correct Incorrect Correct answer

\(x \equiv 5, 6 \pmod{11}\)

Correct Incorrect Correct answer

148. Question

Show that \(x^2 \equiv -1 \pmod{p}\) has a solution if and only if \(p \equiv 1 \pmod{4}\).

When \(p \equiv 3 \pmod{4}\)

Correct Incorrect Correct answer

Always for odd \(p\)

Correct Incorrect Correct answer

Never for prime \(p\)

Correct Incorrect Correct answer

When \(p \equiv 1 \pmod{4}\)

Correct Incorrect Correct answer

149. Question

For \(p = 13\), solve \(x^2 \equiv -1 \pmod{13}\).

\(x \equiv 4, 9\)

Correct Incorrect Correct answer

\(x \equiv 3, 10\)

Correct Incorrect Correct answer

\(x \equiv 5, 8 \pmod{13}\)

Correct Incorrect Correct answer

No solution exists

Correct Incorrect Correct answer

150. Question

Prove that \(x^2 \equiv a \pmod{p}\) has two solutions when \(a\) is a quadratic residue and \(p\) is odd prime.

One

Correct Incorrect Correct answer

Two distinct modulo \(p\)

Correct Incorrect Correct answer

None

Correct Incorrect Correct answer

Three when \(p \equiv 1 \pmod{3}\)

Correct Incorrect Correct answer

151. Question

Find all \(x\) such that \(7x \equiv 7 \pmod{10}\).

\(x \equiv 3 \pmod{10}\)

Correct Incorrect Correct answer

\(x \equiv 7 \pmod{10}\)

Correct Incorrect Correct answer

\(x \equiv 1 \pmod{10}\)

Correct Incorrect Correct answer

\(x \equiv 0 \pmod{10}\)

Correct Incorrect Correct answer

152. Question

Compute \(8^{-1} \pmod{13}\).

\(8^{-1} \equiv 3 \pmod{13}\)

Correct Incorrect Correct answer

\(8^{-1} \equiv 8 \pmod{13}\)

Correct Incorrect Correct answer

\(8^{-1} \equiv 10 \pmod{13}\)

Correct Incorrect Correct answer

\(8^{-1} \equiv 5 \pmod{13}\)

Correct Incorrect Correct answer

153. Question

Evaluate \(30^{17} \pmod{7}\).

\(30^{17} \equiv 1 \pmod{7}\)

Correct Incorrect Correct answer

\(30^{17} \equiv 4 \pmod{7}\)

Correct Incorrect Correct answer

\(30^{17} \equiv 2 \pmod{7}\)

Correct Incorrect Correct answer

\(30^{17} \equiv 6 \pmod{7}\)

Correct Incorrect Correct answer

154. Question

Check whether 68 is a quadratic residue mod 11 by direct computation.

Not a residue

Correct Incorrect Correct answer

Residue, since \(68 \equiv 4 \pmod{11}\)

Correct Incorrect Correct answer

Residue, because 68 is even

Correct Incorrect Correct answer

Residue, because 68 appears in the mod 11 square list

Correct Incorrect Correct answer

155. Question

Compute \(6! \pmod{11}\).

\(6! \equiv 0 \pmod{11}\)

Correct Incorrect Correct answer

\(6! \equiv 5 \pmod{11}\)

Correct Incorrect Correct answer

\(6! \equiv 1 \pmod{11}\)

Correct Incorrect Correct answer

\(6! \equiv 10 \pmod{11}\)

Correct Incorrect Correct answer

156. Question

True or false: \(25^{13} \equiv 25 \pmod{26}\).

False, because \(25^{12} \not\equiv 1 \pmod{26}\)

Correct Incorrect Correct answer

False, because 25 is not invertible mod 26

Correct Incorrect Correct answer

\(25^{13} \equiv 1 \pmod{26}\)

Correct Incorrect Correct answer

True, by Euler: \(25^{12} \equiv 1 \pmod{26} \Rightarrow 25^{13} \equiv 25 \pmod{26}\)

Correct Incorrect Correct answer

157. Question

Find least positive \(x\) such that \(x \equiv 2 \pmod{5}\) and \(x \equiv 3 \pmod{7}\).

\(x = 12\)

Correct Incorrect Correct answer

\(x = 17\)

Correct Incorrect Correct answer

\(x = 24\)

Correct Incorrect Correct answer

\(x = 38\)

Correct Incorrect Correct answer

158. Question

Compute \(11^{100} \pmod{12}\).

\(11^{100} \equiv 11 \pmod{12}\)

Correct Incorrect Correct answer

\(11^{100} \equiv 0 \pmod{12}\)

Correct Incorrect Correct answer

\(11^{100} \equiv 1 \pmod{12}\)

Correct Incorrect Correct answer

\(11^{100} \equiv 5 \pmod{12}\)

Correct Incorrect Correct answer

159. Question

Solve \(9x \equiv 6 \pmod{15}\).

\(x \equiv 3, 6, 9 \pmod{15}\)

Correct Incorrect Correct answer

\(x \equiv 4, 9, 14 \pmod{15}\)

Correct Incorrect Correct answer

\(x \equiv 1, 6, 11 \pmod{15}\)

Correct Incorrect Correct answer

\(x \equiv 2, 5, 8 \pmod{15}\)

Correct Incorrect Correct answer

160. Question

Show that every odd square is congruent to 1 mod 8.

Only if \(n \equiv 3 \pmod{8}\)

Correct Incorrect Correct answer

True only for primes

Correct Incorrect Correct answer

\(n^2 \equiv 0 \pmod{8}\) for odd \(n\)

Correct Incorrect Correct answer

Every odd number \(n = 2k+1\) leads to \(n^2 \equiv 1 \pmod{8}\)

Correct Incorrect Correct answer

161. Question

Solve \(14x \equiv 30 \pmod{44}\).

\(x \equiv 13, 33 \pmod{44}\)

Correct Incorrect Correct answer

\(x \equiv 11, 31 \pmod{44}\)

Correct Incorrect Correct answer

\(x \equiv 21, 43 \pmod{44}\)

Correct Incorrect Correct answer

\(x \equiv 17, 39 \pmod{44}\)

Correct Incorrect Correct answer

162. Question

Find \(x\) such that \(x \equiv 4 \pmod{9}\) and \(x \equiv 5 \pmod{10}\).

\(x = 85\)

Correct Incorrect Correct answer

\(x = 55\)

Correct Incorrect Correct answer

\(x = 95\)

Correct Incorrect Correct answer

\(x = 45\)

Correct Incorrect Correct answer

163. Question

Compute \(7^{123} \pmod{13}\).

\(7^{123} \equiv 6 \pmod{13}\)

Correct Incorrect Correct answer

\(7^{123} \equiv 11 \pmod{13}\)

Correct Incorrect Correct answer

\(7^{123} \equiv 5 \pmod{13}\)

Correct Incorrect Correct answer

\(7^{123} \equiv 9 \pmod{13}\)

Correct Incorrect Correct answer

164. Question

Prove: If \(a \equiv b \pmod{n}\), then \(a^k \equiv b^k \pmod{n}\).

Only for even \(k\)

Correct Incorrect Correct answer

Holds because \(n \mid (a – b)\) implies \(n \mid (a^k – b^k)\)

Correct Incorrect Correct answer

Only true if \(n\) is prime

Correct Incorrect Correct answer

Only true if \(a = b\)

Correct Incorrect Correct answer

165. Question

Solve the system: \(x \equiv 2 \pmod{3}\), \(x \equiv 3 \pmod{4}\), \(x \equiv 2 \pmod{5}\).

\(x = 57\)

Correct Incorrect Correct answer

\(x = 62\)

Correct Incorrect Correct answer

\(x = 47\)

Correct Incorrect Correct answer

\(x = 52\)

Correct Incorrect Correct answer

166. Question

Compute the inverse of \(17 \pmod{43}\).

\(17^{-1} \equiv 5 \pmod{43}\)

Correct Incorrect Correct answer

\(17^{-1} \equiv 9 \pmod{43}\)

Correct Incorrect Correct answer

\(17^{-1} \equiv 38 \pmod{43}\)

Correct Incorrect Correct answer

\(17^{-1} \equiv 15 \pmod{43}\)

Correct Incorrect Correct answer

167. Question

Find the smallest \(n\) such that \(n \equiv 1 \pmod{4}\), \(n \equiv 2 \pmod{7}\), \(n \equiv 3 \pmod{9}\).

\(n = 85\)

Correct Incorrect Correct answer

\(n = 93\)

Correct Incorrect Correct answer

\(n = 65\)

Correct Incorrect Correct answer

\(n = 113\)

Correct Incorrect Correct answer

168. Question

Evaluate \(3^{222} \pmod{37}\).

\(3^{222} \equiv 0 \pmod{37}\)

Correct Incorrect Correct answer

\(3^{222} \equiv 1 \pmod{37}\)

Correct Incorrect Correct answer

\(3^{222} \equiv 11 \pmod{37}\)

Correct Incorrect Correct answer

\(3^{222} \equiv 26 \pmod{37}\)

Correct Incorrect Correct answer

169. Question

Show that if \(p\) is prime and \(a \equiv b \pmod{p}\), then \(a^p \equiv b^p \pmod{p^2}\).

True by binomial expansion: higher terms are divisible by \(p^2\)

Correct Incorrect Correct answer

Only for \(a = b\) exactly

Correct Incorrect Correct answer

Only if \(a\) and \(b\) are even

Correct Incorrect Correct answer

Only valid if \(p = 2\)

Correct Incorrect Correct answer

170. Question

Find all \(x\) such that \(x^2 \equiv 1 \pmod{21}\).

\(x \equiv 1, 10, 13, 20 \pmod{21}\)

Correct Incorrect Correct answer

\(x \equiv 1, 7, 14, 20 \pmod{21}\)

Correct Incorrect Correct answer

\(x \equiv 1, 8, 13, 20 \pmod{21}\)

Correct Incorrect Correct answer

\(x \equiv 1, 9, 11, 20 \pmod{21}\)

Correct Incorrect Correct answer

171. Question

Compute \(49^{-1} \pmod{101}\).

\(49^{-1} \equiv 77 \pmod{101}\)

Correct Incorrect Correct answer

\(49^{-1} \equiv 19 \pmod{101}\)

Correct Incorrect Correct answer

\(49^{-1} \equiv 33 \pmod{101}\)

Correct Incorrect Correct answer

\(49^{-1} \equiv 69 \pmod{101}\)

Correct Incorrect Correct answer

172. Question

Use CRT to solve:
\(x \equiv 4 \pmod{11}\),
\(x \equiv 7 \pmod{13}\),
\(x \equiv 2 \pmod{17}\).

\(x \equiv 1144 \pmod{2431}\)

Correct Incorrect Correct answer

\(x \equiv 1044 \pmod{2431}\)

Correct Incorrect Correct answer

\(x \equiv 1344 \pmod{2431}\)

Correct Incorrect Correct answer

\(x \equiv 1244 \pmod{2431}\)

Correct Incorrect Correct answer

173. Question

Prove cancellation law: If \(\gcd(k,n) = 1\) and \(ka \equiv kb \pmod{n}\), then \(a \equiv b \pmod{n}\).

Only if \(k\) is even

Correct Incorrect Correct answer

Only if \(n\) is prime

Correct Incorrect Correct answer

True, since \(n \mid k(a – b)\) and \(k\) is invertible

Correct Incorrect Correct answer

False, cancellation doesn’t hold in modular arithmetic

Correct Incorrect Correct answer

174. Question

Solve \(6x + 9 \equiv 1 \pmod{13}\).

\(x \equiv 3 \pmod{13}\)

Correct Incorrect Correct answer

\(x \equiv 5 \pmod{13}\)

Correct Incorrect Correct answer

\(x \equiv 2 \pmod{13}\)

Correct Incorrect Correct answer

\(x \equiv 7 \pmod{13}\)

Correct Incorrect Correct answer

175. Question

Compute \(2^{\phi(35)} \pmod{35}\).

\(2^{\phi(35)} \equiv 2 \pmod{35}\)

Correct Incorrect Correct answer

\(2^{\phi(35)} \equiv 1 \pmod{35}\)

Correct Incorrect Correct answer

\(2^{\phi(35)} \equiv 17 \pmod{35}\)

Correct Incorrect Correct answer

\(2^{\phi(35)} \equiv 19 \pmod{35}\)

Correct Incorrect Correct answer

176. Question

Let \(p\) be an odd prime. Show that \(2p\) has exactly one solution to \(x^2 \equiv 1 \pmod{2p}\) besides \(x \equiv \pm 1\).

\(x \equiv p \pmod{2p}\)

Correct Incorrect Correct answer

\(x \equiv p + 1 \pmod{2p}\)

Correct Incorrect Correct answer

No other solution exists except \(\pm 1\)

Correct Incorrect Correct answer

\(x \equiv p + 1 \pmod{2p}\) is a third solution distinct from \(\pm 1\)

Correct Incorrect Correct answer

177. Question

Solve \(15x \equiv 1 \pmod{26}\).

\(x \equiv 15 \pmod{26}\)

Correct Incorrect Correct answer

\(x \equiv 7 \pmod{26}\)

Correct Incorrect Correct answer

\(x \equiv 3 \pmod{26}\)

Correct Incorrect Correct answer

\(x \equiv 11 \pmod{26}\)

Correct Incorrect Correct answer

178. Question

Compute \(10^{1000} \pmod{13}\).

\(10^{1000} \equiv 3 \pmod{13}\)

Correct Incorrect Correct answer

\(10^{1000} \equiv 1 \pmod{13}\)

Correct Incorrect Correct answer

\(10^{1000} \equiv 10 \pmod{13}\)

Correct Incorrect Correct answer

\(10^{1000} \equiv 9 \pmod{13}\)

Correct Incorrect Correct answer

179. Question

Evaluate \(111^{111} \pmod{37}\).

\(111^{111} \equiv 7 \pmod{37}\)

Correct Incorrect Correct answer

\(111^{111} \equiv 1 \pmod{37}\)

Correct Incorrect Correct answer

\(111^{111} \equiv 17 \pmod{37}\)

Correct Incorrect Correct answer

\(111^{111} \equiv 0 \pmod{37}\)

Correct Incorrect Correct answer

180. Question

Solve the system using CRT:
\(x \equiv 1 \pmod{2}\),
\(x \equiv 2 \pmod{3}\),
\(x \equiv 3 \pmod{5}\).

\(x \equiv 29 \pmod{30}\)

Correct Incorrect Correct answer

\(x \equiv 23 \pmod{30}\)

Correct Incorrect Correct answer

\(x \equiv 23 \pmod{30}\)

Correct Incorrect Correct answer

\(x \equiv 28 \pmod{30}\)

Correct Incorrect Correct answer

181. Question

Compute the remainder of \(101! \pmod{103}\).

\(101! \equiv 1 \pmod{103}\)

Correct Incorrect Correct answer

\(101! \equiv 1 \pmod{103}\)

Correct Incorrect Correct answer

\(101! \equiv -1 \pmod{103}\)

Correct Incorrect Correct answer

\(101! \equiv 0 \pmod{103}\)

Correct Incorrect Correct answer

182. Question

Let \(p\) be prime. Show \(a^p \equiv a \pmod{p}\) for any integer \(a\).

True by Fermat’s Little Theorem

Correct Incorrect Correct answer

Only holds if \(a < p\)

Correct Incorrect Correct answer

Only true when \(a\) is odd

Correct Incorrect Correct answer

Only if \(p\) divides \(a\)

Correct Incorrect Correct answer

183. Question

Find \(x\) such that \(x^2 \equiv -1 \pmod{29}\).

No solution exists

Correct Incorrect Correct answer

\(x \equiv 14, 15 \pmod{29}\)

Correct Incorrect Correct answer

\(x \equiv 11, 18 \pmod{29}\)

Correct Incorrect Correct answer

\(x \equiv 12, 17 \pmod{29}\)

Correct Incorrect Correct answer

184. Question

Show that \(x^2 \equiv 2 \pmod{7}\) has no solution.

2 not in the list of quadratic residues mod 7

Correct Incorrect Correct answer

\(x \equiv 3 \pmod{7}\) is a solution

Correct Incorrect Correct answer

\(x \equiv 5 \pmod{7}\) is a solution

Correct Incorrect Correct answer

2 is a square mod 7

Correct Incorrect Correct answer

185. Question

Find \(x\) such that \(x^2 \equiv 4 \pmod{15}\).

\(x \equiv 1, 14 \pmod{15}\)

Correct Incorrect Correct answer

\(x \equiv 2, 13 \pmod{15}\)

Correct Incorrect Correct answer

\(x \equiv 3, 12 \pmod{15}\)

Correct Incorrect Correct answer

\(x \equiv 4, 11 \pmod{15}\)

Correct Incorrect Correct answer

186. Question

Write 46₁₀ in base 5.

\(120_{5}\)

Correct Incorrect Correct answer

\(114_{5}\)

Correct Incorrect Correct answer

\(134_{5}\)

Correct Incorrect Correct answer

\(141_{5}\)

Correct Incorrect Correct answer

187. Question

Convert \(10110_2\) to decimal.

\(22_{10}\)

Correct Incorrect Correct answer

\(26_{10}\)

Correct Incorrect Correct answer

\(18_{10}\)

Correct Incorrect Correct answer

\(30_{10}\)

Correct Incorrect Correct answer

188. Question

Express \(210_3\) in decimal.

\(19_{10}\)

Correct Incorrect Correct answer

\(25_{10}\)

Correct Incorrect Correct answer

\(21_{10}\)

Correct Incorrect Correct answer

\(27_{10}\)

Correct Incorrect Correct answer

189. Question

Convert \(89_{10}\) to hexadecimal.

\(0x5A\)

Correct Incorrect Correct answer

\(0x4F\)

Correct Incorrect Correct answer

\(0x59\)

Correct Incorrect Correct answer

\(0x69\)

Correct Incorrect Correct answer

190. Question

Write \(0.625_{10}\) in binary (exact).

\(0.110_2\)

Correct Incorrect Correct answer

\(0.101_2\)

Correct Incorrect Correct answer

\(0.111_2\)

Correct Incorrect Correct answer

\(0.100_2\)

Correct Incorrect Correct answer

191. Question

Which base-10 number has the base-7 representation \(125_7\)?

\(68_{10}\)

Correct Incorrect Correct answer

\(72_{10}\)

Correct Incorrect Correct answer

\(74_{10}\)

Correct Incorrect Correct answer

\(65_{10}\)

Correct Incorrect Correct answer

192. Question

What is the radix of a system whose digits run from 0 to 6?

\(5\)

Correct Incorrect Correct answer

\(7\)

Correct Incorrect Correct answer

\(8\)

Correct Incorrect Correct answer

\(6\)

Correct Incorrect Correct answer

193. Question

Explain why no single digit in base 10 can equal 12.

Because 12 is not a valid digit in base 10; maximum digit is 10.

Correct Incorrect Correct answer

Base 10 digits go only from 1 to 10.

Correct Incorrect Correct answer

Base 10 digits stop at 11, so 12 is excluded.

Correct Incorrect Correct answer

Base 10 digits range from 0 to 9, so 12 requires two digits.

Correct Incorrect Correct answer

194. Question

Identify the weight of the 7 in \(3.7_8\).

\(0.875_8\)

Correct Incorrect Correct answer

\(0.875_{2}\)

Correct Incorrect Correct answer

\(0.875_{10}\)

Correct Incorrect Correct answer

\(0.875_{16}\)

Correct Incorrect Correct answer

195. Question

What is the place value of the digit 3 in \(732.4_8\)?

\(96_{10}\)

Correct Incorrect Correct answer

\(192_{10}\)

Correct Incorrect Correct answer

\(128_{10}\)

Correct Incorrect Correct answer

\(256_{10}\)

Correct Incorrect Correct answer

196. Question

Compute \(1001^{-1} \mod 1013\).

\(591\)

Correct Incorrect Correct answer

\(422\)

Correct Incorrect Correct answer

\(912\)

Correct Incorrect Correct answer

\(591 \equiv -422 \mod 1013\)

Correct Incorrect Correct answer

197. Question

Find the least positive \(x\) such that
\(x \equiv 3 \pmod{11}, \quad x \equiv 4 \pmod{12}, \quad x \equiv 5 \pmod{13}\)

\(x = 1716\)

Correct Incorrect Correct answer

\(x = 1689\)

Correct Incorrect Correct answer

\(x = 1225\)

Correct Incorrect Correct answer

\(x = 1803\)

Correct Incorrect Correct answer

198. Question

Compute \(7^{\phi(30) + 1} \mod 30\).

\(49\)

Correct Incorrect Correct answer

\(1\)

Correct Incorrect Correct answer

\(7\)

Correct Incorrect Correct answer

\(19\)

Correct Incorrect Correct answer

199. Question

Use the digit-sum trick to find the remainder of \(123456789 \mod 9\).

\(4\)

Correct Incorrect Correct answer

\(5\)

Correct Incorrect Correct answer

\(0\)

Correct Incorrect Correct answer

\(3\)

Correct Incorrect Correct answer

200. Question

Solve \(28x \equiv 37 \mod 55\).

\(x = 17\)

Correct Incorrect Correct answer

\(x = 19\)

Correct Incorrect Correct answer

\(x = 23\)

Correct Incorrect Correct answer

\(x = 21\)

Correct Incorrect Correct answer

201. Question

Compute \(3^{2005} \mod 7\).

\(1\)

Correct Incorrect Correct answer

\(4\)

Correct Incorrect Correct answer

\(3\)

Correct Incorrect Correct answer

\(6\)

Correct Incorrect Correct answer

202. Question

Let \(p = 11\). Show \(a^{10} \equiv 1 \mod p\) for \(a = 2, 3, …, 10\).

Follows from Fermat’s Little Theorem (FLT): \(a^{p-1} \equiv 1 \mod p\)

Correct Incorrect Correct answer

Only holds for even numbers \(a\)

Correct Incorrect Correct answer

True only when \(a\) is a prime

Correct Incorrect Correct answer

Works only when \(a \lt p/2\)

Correct Incorrect Correct answer

203. Question

Prove that if \(m \mid n\) then \(a \equiv b \mod n \Rightarrow a \equiv b \mod m\).

Since \(m \mid n\), then \(m \mid (a – b)\)

Correct Incorrect Correct answer

\(n \mid a \Rightarrow m \mid b\)

Correct Incorrect Correct answer

\(n \mid (a + b)\)

Correct Incorrect Correct answer

\(a \equiv b \mod m\) implies \(m \mid a\)

Correct Incorrect Correct answer

204. Question

Find all \(x\) such that \(x^2 \equiv 4 \mod 15\).

\(x \equiv 2, 5, 10, 13 \mod 15\)

Correct Incorrect Correct answer

\(x \equiv 3, 6, 9, 12 \mod 15\)

Correct Incorrect Correct answer

\(x \equiv 0, 5, 10, 15 \mod 15\)

Correct Incorrect Correct answer

\(x \equiv 1, 4, 11, 14 \mod 15\)

Correct Incorrect Correct answer

205. Question

Show that \(9 \mid (a^3 – a)\) for all integers \(a\).

\(a^3 – a\) is always even

Correct Incorrect Correct answer

Because \(a^3 \equiv a \mod 3\), so \(a^3 – a\) divisible by 3 and thus by 9

Correct Incorrect Correct answer

\(a \mod 9 = a^3 \mod 9\)

Correct Incorrect Correct answer

Only true when \(a\) is divisible by 3

Correct Incorrect Correct answer

206. Question

Solve \(x \equiv 1 \mod 2, \quad x \equiv 2 \mod 3, \quad x \equiv 3 \mod 5\).

\(x = 27\)

Correct Incorrect Correct answer

\(x = 33\)

Correct Incorrect Correct answer

\(x = 23\)

Correct Incorrect Correct answer

\(x = 18\)

Correct Incorrect Correct answer

207. Question

Prove that \(a \equiv b \mod n\) iff \(a\) and \(b\) have the same remainder on division by \(n\).

True only if \(n \mid a\)

Correct Incorrect Correct answer

Their difference \(a – b\) is divisible by \(n\)

Correct Incorrect Correct answer

Only true for even \(n\)

Correct Incorrect Correct answer

Only true if \(a < n\)

Correct Incorrect Correct answer

208. Question

Show there are infinitely many primes \(p\) such that \(p \equiv 1 \mod 4\).

Use FLT with \(p = 4k + 1\) and derive contradiction

Correct Incorrect Correct answer

Use Dirichlet’s Theorem on arithmetic progressions

Correct Incorrect Correct answer

Fermat’s Last Theorem proves this

Correct Incorrect Correct answer

Use Wilson’s Theorem on \(p \equiv 1 \mod 4\)

Correct Incorrect Correct answer

209. Question

Prove Wilson’s theorem for \(p = 7\): \(6! \equiv -1 \mod 7\).

\(6! = 720, 720 \mod 7 = 0\)

Correct Incorrect Correct answer

\(6! = 720, 720 \mod 7 = 3\)

Correct Incorrect Correct answer

\(6! = 720, 720 \equiv -1 \mod 7\)

Correct Incorrect Correct answer

\(6! \equiv 1 \mod 7\)

Correct Incorrect Correct answer

210. Question

Let \(n = 2^k\). Prove that units mod \(n\) are odd numbers.

Odds are coprime to \(2^k\), evens share factor 2

Correct Incorrect Correct answer

Even numbers are coprime to \(2^k\)

Correct Incorrect Correct answer

All numbers \(< n\) are units

Correct Incorrect Correct answer

Only even numbers are invertible

Correct Incorrect Correct answer

211. Question

Find \(x\) with \(0 < x < 77\) such that \(x \equiv 3 \mod 11\) and \(x \equiv 4 \mod 7\).

\(x = 69\)

Correct Incorrect Correct answer

\(x = 36\)

Correct Incorrect Correct answer

\(x = 47\)

Correct Incorrect Correct answer

\(x = 58\)

Correct Incorrect Correct answer

212. Question

Compute \(2^{1000} \mod 125\).

\(0\)

Correct Incorrect Correct answer

\(76\)

Correct Incorrect Correct answer

\(100\)

Correct Incorrect Correct answer

\(25\)

Correct Incorrect Correct answer

213. Question

Show that \(7 \mid \binom{2k}{k}\) for all \(k > 1\).

Use Lucas’s theorem; binomial divisible by 7 for \(k > 1\)

Correct Incorrect Correct answer

Binomial is always even, hence divisible by 7

Correct Incorrect Correct answer

Fermat’s theorem implies it

Correct Incorrect Correct answer

Follows from Euler’s criterion

Correct Incorrect Correct answer

214. Question

Prove CRT (Chinese Remainder Theorem) by constructing explicit inverse modulo \(n_i\).

Use repeated squaring and Wilson’s theorem

Correct Incorrect Correct answer

Let \(x = \prod n_i\) and apply Euler’s theorem

Correct Incorrect Correct answer

Let \(N_i = N/n_i\), find \(e_i \equiv N_i^{-1} \mod n_i\), then \(x = \sum a_i N_i e_i\)

Correct Incorrect Correct answer

All values \(x\) are congruent to 0 mod \(N\)

Correct Incorrect Correct answer

215. Question

Find all primes \(p\) such that \(p \mid (2^p – 2)\).

All primes \(p > 2\)

Correct Incorrect Correct answer

Only \(p = 2, 3\)

Correct Incorrect Correct answer

Wieferich primes base 2: \(p = 1093, 3511\)

Correct Incorrect Correct answer

No such primes exist

Correct Incorrect Correct answer

216. Question

Show \(x^{\phi(n)+1} \equiv x \mod n\) for \(\gcd(x, n) = 1\).

Fermat’s Little Theorem: \(x^p \equiv x \mod p\)

Correct Incorrect Correct answer

Only true when \(x\) is prime

Correct Incorrect Correct answer

Only valid if \(n\) is a power of 2

Correct Incorrect Correct answer

Euler’s theorem: \(x^{\phi(n)} \equiv 1 \Rightarrow x^{\phi(n)+1} \equiv x\)

Correct Incorrect Correct answer

217. Question

Solve \(x^2 \equiv 2 \mod 13\).

\(x = 5, 8\)

Correct Incorrect Correct answer

\(x = 4, 9\)

Correct Incorrect Correct answer

\(x = 6, 7\)

Correct Incorrect Correct answer

No solution exists

Correct Incorrect Correct answer

218. Question

Prove \(\sum_{k=1}^{p-1} k^{p-1} \equiv -1 \mod p\) for prime \(p\).

Sum equals \(p – 1\) mod \(p\)

Correct Incorrect Correct answer

Each term \(\equiv 1\), sum is \(p – 1 \equiv -1 \mod p\)

Correct Incorrect Correct answer

All terms cancel to 0

Correct Incorrect Correct answer

Sum always divisible by \(p^2\)

Correct Incorrect Correct answer

219. Question

Show that for odd \(n\), \(n^2 \equiv 1 \mod 8\).

Follows from square of any odd number being \(\equiv 1 \mod 8\)

Correct Incorrect Correct answer

Only true for \(n \equiv 1 \mod 8\)

Correct Incorrect Correct answer

Even \(n\) also satisfy this

Correct Incorrect Correct answer

Only holds if \(n\) is prime

Correct Incorrect Correct answer

220. Question

Find inverse of \(3 \mod 97\) and solve \(3x \equiv 25 \mod 97\).

Inverse is 32, so \(x = 64\)

Correct Incorrect Correct answer

Inverse is 75, so \(x = 45\)

Correct Incorrect Correct answer

Inverse is 65, so \(x = 65 \times 25 = 1625 \equiv 73 \mod 97\)

Correct Incorrect Correct answer

Inverse is 25, so \(x = 1625 \equiv 26 \mod 97\)

Correct Incorrect Correct answer

221. Question

If \(a \equiv b \mod n\) and \(c \equiv d \mod n\), then \(ac \equiv bd \mod n\).

Only true if \(a = c\) and \(b = d\)

Correct Incorrect Correct answer

Follows since \((a – b)(c – d)\) is divisible by \(n\)

Correct Incorrect Correct answer

True when \(a, b, c, d\) are all less than \(n\)

Correct Incorrect Correct answer

Holds only if \(n\) is prime

Correct Incorrect Correct answer

222. Question

Find the smallest positive \(x\) such that:

\(x = 58\)

Correct Incorrect Correct answer

\(x = 135\)

Correct Incorrect Correct answer

\(x = 73\)

Correct Incorrect Correct answer

\(x = 73\) (from correct CRT solution)

Correct Incorrect Correct answer

223. Question

Compute \(13^{45} \mod 37\).

\(13\)

Correct Incorrect Correct answer

\(24\)

Correct Incorrect Correct answer

\(7\)

Correct Incorrect Correct answer

\(6\)

Correct Incorrect Correct answer

224. Question

Show that the product of all units mod prime \(p\) is \(-1\).

Pair each unit \(a\) with its inverse \(a^{-1}\), except \(1\) and \(p – 1\)

Correct Incorrect Correct answer

Product is \(1\) because all units cancel

Correct Incorrect Correct answer

Product always \(0\) mod \(p\)

Correct Incorrect Correct answer

Only works if \(p\) is even

Correct Incorrect Correct answer

225. Question

Prove: If \(a \mid bc\) and \(\gcd(a, b) = 1\), then \(a \mid c\).

Use Wilson’s theorem

Correct Incorrect Correct answer

Since \(a \mid b\) and \(b \mid c\)

Correct Incorrect Correct answer

Use Bézout: \(au + bv = 1 \Rightarrow a \mid c\) from \(c = u(bc) + v(ac)\)

Correct Incorrect Correct answer

Only true if \(b \mid a\)

Correct Incorrect Correct answer

226. Question

Show that \(\binom{2n}{n}\) is even for all \(n \geq 1\).

Follows from Lucas’s Theorem

Correct Incorrect Correct answer

Central binomial coefficient divisible by 2 due to factorial powers

Correct Incorrect Correct answer

Always odd for odd \(n\)

Correct Incorrect Correct answer

Only even when \(n\) is prime

Correct Incorrect Correct answer

227. Question

RSA Toy Setup: Choose \(p = 11\), \(q = 17\). Compute \(n\) and \(\varphi(n)\).

\(n = 187, \varphi(n) = 180\)

Correct Incorrect Correct answer

\(n = 187, \varphi(n) = 160\)

Correct Incorrect Correct answer

\(n = 176, \varphi(n) = 150\)

Correct Incorrect Correct answer

\(n = 187, \varphi(n) = 176\)

Correct Incorrect Correct answer

228. Question

Select public exponent \(e = 7\). Find private key \(d\) such that \(7d \equiv 1 \mod 160\).

\(d = 27\)

Correct Incorrect Correct answer

\(d = 17\)

Correct Incorrect Correct answer

\(d = 33\)

Correct Incorrect Correct answer

\(d = 23\)

Correct Incorrect Correct answer

229. Question

Encrypt message \(m = 100\) using \(e = 7\), \(n = 187\).

\(100^7 \mod 187 = 41\)

Correct Incorrect Correct answer

\(100^7 \mod 187 = 91\)

Correct Incorrect Correct answer

\(100^7 \mod 187 = 53\)

Correct Incorrect Correct answer

\(100^7 \mod 187 = 77\)

Correct Incorrect Correct answer

230. Question

Decrypt cipher \(c = 53\) using private key \(d = 23\) and \(n = 187\).

\(53^{23} \mod 187 = 91\)

Correct Incorrect Correct answer

\(53^{23} \mod 187 = 123\)

Correct Incorrect Correct answer

\(53^{23} \mod 187 = 115\)

Correct Incorrect Correct answer

\(53^{23} \mod 187 = 100\)

Correct Incorrect Correct answer

231. Question

Write 46₁₀ in base 5.

\(141_5\)

Correct Incorrect Correct answer

\(114_5\)

Correct Incorrect Correct answer

\(144_5\)

Correct Incorrect Correct answer

\(411_5\)

Correct Incorrect Correct answer

232. Question

Convert \(10110_2\) to decimal.

\(21_{10}\)

Correct Incorrect Correct answer

\(24_{10}\)

Correct Incorrect Correct answer

\(22_{10}\)

Correct Incorrect Correct answer

\(18_{10}\)

Correct Incorrect Correct answer

233. Question

What is the place value of the digit 3 in \(732.4_8\)?

\(192_{10}\)

Correct Incorrect Correct answer

\(64_{10}\)

Correct Incorrect Correct answer

\(24_{10}\)

Correct Incorrect Correct answer

\(384_{10}\)

Correct Incorrect Correct answer

234. Question

Express \(210_3\) in decimal.

\(23_{10}\)

Correct Incorrect Correct answer

\(21_{10}\)

Correct Incorrect Correct answer

\(19_{10}\)

Correct Incorrect Correct answer

\(13_{10}\)

Correct Incorrect Correct answer

235. Question

Explain why no single digit in base 10 can equal 12.

Because base 10 only has digits from 0–9, so 12 requires two digits

Correct Incorrect Correct answer

Because base 10 skips 12 altogether

Correct Incorrect Correct answer

Because 12 is not a number in base 10

Correct Incorrect Correct answer

Because 10 + 2 = 12 in base 10, so it’s single digit

Correct Incorrect Correct answer

236. Question

Identify the weight of the 7 in \(3.7_8\).

\(7.875_{10}\)

Correct Incorrect Correct answer

\(0.0875_{10}\)

Correct Incorrect Correct answer

\(1.75_{10}\)

Correct Incorrect Correct answer

\(0.875_{10}\)

Correct Incorrect Correct answer

237. Question

Convert \(3C_{16}\) to decimal.

\(51_{10}\)

Correct Incorrect Correct answer

\(45_{10}\)

Correct Incorrect Correct answer

\(60_{10}\)

Correct Incorrect Correct answer

\(72_{10}\)

Correct Incorrect Correct answer

238. Question

What base has 100 representing \(9_{10}\)?

\(9\)

Correct Incorrect Correct answer

\(6\)

Correct Incorrect Correct answer

\(2\)

Correct Incorrect Correct answer

\(3\)

Correct Incorrect Correct answer

239. Question

Evaluate \(0.1_3\) in decimal.

\(0.3\)

Correct Incorrect Correct answer

\(0.333\ldots\)

Correct Incorrect Correct answer

\(0.13\)

Correct Incorrect Correct answer

\(0.23\)

Correct Incorrect Correct answer

240. Question

Write \(78.8_{16}\) in decimal (to 3 dp).

\(123.8\)

Correct Incorrect Correct answer

\(127.5\)

Correct Incorrect Correct answer

\(123.5\)

Correct Incorrect Correct answer

\(129.8\)

Correct Incorrect Correct answer

241. Question

Convert \(640_7\) to base 5.

\(2422_5\)

Correct Incorrect Correct answer

\(4322_5\)

Correct Incorrect Correct answer

\(3422_5\)

Correct Incorrect Correct answer

\(2242_5\)

Correct Incorrect Correct answer

242. Question

Show \(0.1_2\) is repeating in base 10.

False – it terminates

Correct Incorrect Correct answer

True – it repeats

Correct Incorrect Correct answer

True – it never ends

Correct Incorrect Correct answer

False – it’s irrational

Correct Incorrect Correct answer

243. Question

Express \(2AF_{16}\) in base 2.

\(101010101111_2\)

Correct Incorrect Correct answer

\(001010101111_2\)

Correct Incorrect Correct answer

\(101001011111_2\)

Correct Incorrect Correct answer

\(001110101111_2\)

Correct Incorrect Correct answer

244. Question

Find the smallest base \(b\) in which the numeral \(321_b\) is valid and represents an even number.

\(b = 2\)

Correct Incorrect Correct answer

\(b = 3\)

Correct Incorrect Correct answer

Impossible – always odd

Correct Incorrect Correct answer

\(b = 6\)

Correct Incorrect Correct answer

245. Question

Convert \(0.2_3\) to base 9.

\(0.6_9\)

Correct Incorrect Correct answer

\(0.2_9\)

Correct Incorrect Correct answer

\(0.9_9\)

Correct Incorrect Correct answer

\(0.3_9\)

Correct Incorrect Correct answer

246. Question

Convert \(ABC_{16}\) to base 13 (show decimal bridge).

\(1435_{13}\)

Correct Incorrect Correct answer

\(1335_{13}\)

Correct Incorrect Correct answer

\(1535_{13}\)

Correct Incorrect Correct answer

\(1334_{13}\)

Correct Incorrect Correct answer

247. Question

Prove \(n\) has same digits in base \(b\) and \(b – 1\) only if \(n < b\).

Always true for all \(n\)

Correct Incorrect Correct answer

Only true if \(b\) is even

Correct Incorrect Correct answer

Only true for \(n > b\)

Correct Incorrect Correct answer

Only possible when \(n < b\)

Correct Incorrect Correct answer

248. Question

Convert \(0.34_{10}\) to base 2 (6 bits).

\(0.010001_2\)

Correct Incorrect Correct answer

\(0.011001_2\)

Correct Incorrect Correct answer

\(0.010101_2\)

Correct Incorrect Correct answer

\(0.001001_2\)

Correct Incorrect Correct answer

249. Question

Write \(555_6\) in base 30.

\(25_{30}\)

Correct Incorrect Correct answer

\(57_{30}\)

Correct Incorrect Correct answer

\(75_{30}\)

Correct Incorrect Correct answer

\(17_{30}\)

Correct Incorrect Correct answer

250. Question

Evaluate \(1.23_4\) in decimal.

\(1.6875\)

Correct Incorrect Correct answer

\(1.625\)

Correct Incorrect Correct answer

\(1.75\)

Correct Incorrect Correct answer

\(1.375\)

Correct Incorrect Correct answer

251. Question

Change \(458_{10}\) to base 8.

\(721_8\)

Correct Incorrect Correct answer

\(271_8\)

Correct Incorrect Correct answer

\(127_8\)

Correct Incorrect Correct answer

\(712_8\)

Correct Incorrect Correct answer

252. Question

Convert \(17.25_{10}\) to binary.

\(10010.01_2\)

Correct Incorrect Correct answer

\(10001.10_2\)

Correct Incorrect Correct answer

\(10001.01_2\)

Correct Incorrect Correct answer

\(10000.11_2\)

Correct Incorrect Correct answer

253. Question

Write \(7B.8_{16}\) in decimal (to 3 dp).

\(127.5\)

Correct Incorrect Correct answer

\(123.8\)

Correct Incorrect Correct answer

\(120.5\)

Correct Incorrect Correct answer

\(123.5\)

Correct Incorrect Correct answer

254. Question

Convert \(640_7\) to base 5.

\(2422_5\)

Correct Incorrect Correct answer

\(2242_5\)

Correct Incorrect Correct answer

\(4222_5\)

Correct Incorrect Correct answer

\(2224_5\)

Correct Incorrect Correct answer

255. Question

Show \(0.1_2\) is repeating in base 10.

False – it terminates

Correct Incorrect Correct answer

True – it repeats

Correct Incorrect Correct answer

False – it’s irrational

Correct Incorrect Correct answer

True – it terminates

Correct Incorrect Correct answer

256. Question

Convert \(14.1_8\) to binary.

\(1100.010_2\)

Correct Incorrect Correct answer

\(001100.001_2\)

Correct Incorrect Correct answer

\(011100.001_2\)

Correct Incorrect Correct answer

\(001100.010_2\)

Correct Incorrect Correct answer

257. Question

Write \(22_{10}\) as a two-digit number in base \(b\) if possible.

\(22_{11}\)

Correct Incorrect Correct answer

\(11_{11}\)

Correct Incorrect Correct answer

\(12_{11}\)

Correct Incorrect Correct answer

\(20_{11}\)

Correct Incorrect Correct answer

258. Question

Convert \(0.2_3\) to base 9.

\(0.6_9\)

Correct Incorrect Correct answer

\(0.2_9\)

Correct Incorrect Correct answer

\(0.3_9\)

Correct Incorrect Correct answer

\(0.8_9\)

Correct Incorrect Correct answer

259. Question

Convert \(ABC_{16}\) to base 13.

\(1345_{13}\)

Correct Incorrect Correct answer

\(1435_{13}\)

Correct Incorrect Correct answer

\(1335_{13}\)

Correct Incorrect Correct answer

\(1533_{13}\)

Correct Incorrect Correct answer

260. Question

Convert \(0.3_{10}\) to base 2 (6 bits).

\(0.011001_2\)

Correct Incorrect Correct answer

\(0.110001_2\)

Correct Incorrect Correct answer

\(0.101001_2\)

Correct Incorrect Correct answer

\(0.010001_2\)

Correct Incorrect Correct answer

261. Question

Convert 24-hour time “23:45” (base 60) to pure seconds.

\(84600 \text{ seconds}\)

Correct Incorrect Correct answer

\(87500 \text{ seconds}\)

Correct Incorrect Correct answer

\(85500 \text{ seconds}\)

Correct Incorrect Correct answer

\(83500 \text{ seconds}\)

Correct Incorrect Correct answer

262. Question

Identify next palindrome after \(101_2\) in binary.

\(111_2\)

Correct Incorrect Correct answer

\(1001_2\)

Correct Incorrect Correct answer

\(1101_2\)

Correct Incorrect Correct answer

\(1010_2\)

Correct Incorrect Correct answer

263. Question

Convert \(1E.4_{16}\) to base 8.

\(174.0_8\)

Correct Incorrect Correct answer

\(170.4_8\)

Correct Incorrect Correct answer

\(160.4_8\)

Correct Incorrect Correct answer

\(171.4_8\)

Correct Incorrect Correct answer

264. Question

What is the ratio of weights between successive positions in base \(b\)?

\(b:1\)

Correct Incorrect Correct answer

1:b

Correct Incorrect Correct answer

\(b^2:1\)

Correct Incorrect Correct answer

\(1:2b\)

Correct Incorrect Correct answer

265. Question

Convert \(0.125_{10}\) to base 8.

\(0.2_8\)

Correct Incorrect Correct answer

\(0.01_8\)

Correct Incorrect Correct answer

\(0.8_8\)

Correct Incorrect Correct answer

\(0.1_8\)

Correct Incorrect Correct answer

266. Question

Show if base \(b\) is a square, any terminating representation in \(\sqrt{b}\) converts to at most double length.

Every digit becomes four digits

Correct Incorrect Correct answer

Each digit becomes at most two \(\sqrt{b}\)-digits

Correct Incorrect Correct answer

The representation remains the same

Correct Incorrect Correct answer

Digits triple in size

Correct Incorrect Correct answer

267. Question

Find smallest \(n\) where \(10^n\) is 1 followed solely by zeros in base 3.

\(n = 3\)

Correct Incorrect Correct answer

\(n = 2\)

Correct Incorrect Correct answer

\(n = 4\)

Correct Incorrect Correct answer

\(n = 1\)

Correct Incorrect Correct answer

268. Question

Prove a rational number has purely periodic representation in base \(b\) iff value < 1 and denominator coprime to \(b\).

Denominator must divide base

Correct Incorrect Correct answer

Only works when numerator > 1

Correct Incorrect Correct answer

Always finite representation

Correct Incorrect Correct answer

Coprime with base and value < 1 ⇒ purely periodic

Correct Incorrect Correct answer

269. Question

Investigate number of trailing zeros in \(n!\) in base \(b\).

Count digits of \(n!\)

Correct Incorrect Correct answer

Multiply \(n!\) by \(b\)

Correct Incorrect Correct answer

Use minimum sum of floors over all \(p\) dividing \(b\)

Correct Incorrect Correct answer

Divide \(n!\) by 10

Correct Incorrect Correct answer

270. Question

Convert \(19.6_{10}\) to base 6 exactly.

\(31.3_6\)

Correct Incorrect Correct answer

\(31.\overline{3}_6\)

Correct Incorrect Correct answer

\(33.3_6\)

Correct Incorrect Correct answer

\(31.6_6\)

Correct Incorrect Correct answer

271. Question

Evaluate 1.23₄ in decimal.

\(1.625\)

Correct Incorrect Correct answer

\(1.875\)

Correct Incorrect Correct answer

\(1.6875\)

Correct Incorrect Correct answer

\(1.75\)

Correct Incorrect Correct answer

272. Question

Show any terminating decimal has repeating binary or not?

Denominator has factor of \(3\)

Correct Incorrect Correct answer

Denominator has form \(2^a 5^b\); if \(b > 0\), it terminates in binary

Correct Incorrect Correct answer

Denominator has form \(2^a 5^b\); if \(b > 0\), binary repeats

Correct Incorrect Correct answer

Denominator has only factor \(2\); binary terminates

Correct Incorrect Correct answer

273. Question

Convert 89₁₀ to base 3.

\(10022_3\)

Correct Incorrect Correct answer

\(12012_3\)

Correct Incorrect Correct answer

\(10122_3\)

Correct Incorrect Correct answer

\(11022_3\)

Correct Incorrect Correct answer

274. Question

Determine base \(b\) if \(211_b = 65_{10}\)

\(b = 5.417\), acceptable

Correct Incorrect Correct answer

\(b = 8\), verified

Correct Incorrect Correct answer

\(b = 4\), valid

Correct Incorrect Correct answer

No such integer \(b\) exists

Correct Incorrect Correct answer

275. Question

Show \(0.\overline{1}_b = \frac{1}{b-1}\)

\(\frac{b}{b+1}\)

Correct Incorrect Correct answer

\(\frac{1}{b-1}\)

Correct Incorrect Correct answer

\(\frac{1}{b+1}\)

Correct Incorrect Correct answer

\(\frac{b-1}{b}\)

Correct Incorrect Correct answer

276. Question

Convert 24-hour time “23:45” (base 60) to pure seconds.

\(83400\)

Correct Incorrect Correct answer

\(85500\)

Correct Incorrect Correct answer

\(86400\)

Correct Incorrect Correct answer

\(88200\)

Correct Incorrect Correct answer

277. Question

Identify next palindrome after \(101_2\), in binary.

\(1001_2\)

Correct Incorrect Correct answer

\(1101_2\)

Correct Incorrect Correct answer

\(111_2\)

Correct Incorrect Correct answer

\(1010_2\)

Correct Incorrect Correct answer

278. Question

Convert \(1E.4_{16}\) to base 8.

\(170.4_8\)

Correct Incorrect Correct answer

\(160.2_8\)

Correct Incorrect Correct answer

\(172.4_8\)

Correct Incorrect Correct answer

\(161.6_8\)

Correct Incorrect Correct answer

279. Question

Give ratio of weights between successive positions in base \(b\).

Each left position multiplies by \(b^2\)

Correct Incorrect Correct answer

Each left step multiplies by \(b+1\)

Correct Incorrect Correct answer

Each left step multiplies by \(b\)

Correct Incorrect Correct answer

Each left step multiplies by \(2b\)

Correct Incorrect Correct answer

280. Question

Convert \(0.125_{10}\) to base 8.

\(0.2_8\)

Correct Incorrect Correct answer

\(0.1_8\)

Correct Incorrect Correct answer

\(0.12_8\)

Correct Incorrect Correct answer

\(0.01_8\)

Correct Incorrect Correct answer

281. Question

Prove an integer is divisible by 3 in any base if digit sum ≡ 0 (mod 3).

Because \(b \equiv 0 \pmod{3}\)

Correct Incorrect Correct answer

Because digits cancel

Correct Incorrect Correct answer

Because \(b\) is even

Correct Incorrect Correct answer

Because \(b^k \equiv 1 \pmod{3}\)

Correct Incorrect Correct answer

282. Question

Show \(0.2_{10}\) cannot terminate in base 3.

2 is not a factor of 3

Correct Incorrect Correct answer

Prime 5 divides 3

Correct Incorrect Correct answer

Denominator has 5, which base 3 can’t terminate

Correct Incorrect Correct answer

3 is not a prime

Correct Incorrect Correct answer

283. Question

Prove maximum repeating period of \(1/q\) in base \(b \leq q – 1\).

There are \(q – 1\) possible remainders

Correct Incorrect Correct answer

Maximum digits in \(b\)-expansion is \(q + 1\)

Correct Incorrect Correct answer

There are \(q – 1\) digits in total

Correct Incorrect Correct answer

All repeating digits must be odd

Correct Incorrect Correct answer

284. Question

Design algorithm to detect palindromes in base \(b\).

Check if the sum of digits is even

Correct Incorrect Correct answer

Check if digit order matches when reversed

Correct Incorrect Correct answer

Multiply digits by base

Correct Incorrect Correct answer

Use square root comparison

Correct Incorrect Correct answer

285. Question

Convert \(0.\overline{13}_{10}\) to base 7 with 5 digits.

\(0.04206_7\)

Correct Incorrect Correct answer

\(0.06014_7\)

Correct Incorrect Correct answer

\(0.06104_7\)

Correct Incorrect Correct answer

\(0.06710_7\)

Correct Incorrect Correct answer

286. Question

Show if base \(b\) is a square, any terminating representation in \(\sqrt{b}\) converts to at most double length.

Each digit becomes \(b\) digits

Correct Incorrect Correct answer

Each digit stays same

Correct Incorrect Correct answer

Each digit multiplies by \(\sqrt{b}\)

Correct Incorrect Correct answer

Each digit splits into at most two \(\sqrt{b}\) digits

Correct Incorrect Correct answer

287. Question

Find smallest \(n\) where \(10^n\) is 1 followed by only zeros in base 3.

\(n = 3\)

Correct Incorrect Correct answer

\(n = 5\)

Correct Incorrect Correct answer

\(n = 4\)

Correct Incorrect Correct answer

\(n = 6\)

Correct Incorrect Correct answer

288. Question

Prove a rational has purely periodic representation in base \(b\) iff value \(< 1\) and denominator coprime to \(b\).

Denominator is coprime to \(b\)

Correct Incorrect Correct answer

Denominator must be divisible by \(b\)

Correct Incorrect Correct answer

Denominator is a power of \(b\)

Correct Incorrect Correct answer

Numerator is greater than denominator

Correct Incorrect Correct answer

289. Question

Investigate number of trailing zeros of \(n!\) in base \(b\).

Count multiples of \(b\) only

Correct Incorrect Correct answer

Count \(\min_p \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \ldots\) where \(p\) is a prime in \(b\)

Correct Incorrect Correct answer

Count all digits divisible by \(b\)

Correct Incorrect Correct answer

Use remainder after division by \(b\)

Correct Incorrect Correct answer

290. Question

Convert \(19.6_{10}\) to base 6 exactly.

\(31.5_6\)

Correct Incorrect Correct answer

\(30.4_6\)

Correct Incorrect Correct answer

\(32.1_6\)

Correct Incorrect Correct answer

\(31.3̅_6\)

Correct Incorrect Correct answer

291. Question

Show base change from \(b\) to \(b^k\) scales number of digits by \(\frac{1}{k}\) (ceiling).

\(b^{m-1} \leq N < b^m \Rightarrow \text{digits} \approx \frac{m}{k}\)

Correct Incorrect Correct answer

Digits increase by \(k^2\)

Correct Incorrect Correct answer

\(b^{k+1} \approx N^k\)

Correct Incorrect Correct answer

No change in digit count

Correct Incorrect Correct answer

292. Question

Prove primes 2 and 5 control decimal termination.

Terminating when numerator is 2 or 5

Correct Incorrect Correct answer

Decimal must be rational

Correct Incorrect Correct answer

Denominator divisible by only 2 and 5

Correct Incorrect Correct answer

Denominator must be even

Correct Incorrect Correct answer

293. Question

Convert repeating binary \(0.\overline{01}_2\) to decimal.

\(0.25\)

Correct Incorrect Correct answer

\(0.333\)

Correct Incorrect Correct answer

\(0.5\)

Correct Incorrect Correct answer

\(0.666\)

Correct Incorrect Correct answer

294. Question

Show \(1/7\) has period 6 in base 10.

Because \(7\) is prime

Correct Incorrect Correct answer

Because \(7 = 10 – 3\)

Correct Incorrect Correct answer

Because \(1/7\) terminates

Correct Incorrect Correct answer

Because 10 is primitive root mod 7

Correct Incorrect Correct answer

295. Question

Convert \(1000_2\) to base \(b\) formulaically.

\(2^3 = 6\)

Correct Incorrect Correct answer

\(2^4 = 16\)

Correct Incorrect Correct answer

\(2^3 = 8\)

Correct Incorrect Correct answer

\(2^2 = 4\)

Correct Incorrect Correct answer

296. Question

Show base-phi redundant (non-integer) is still positional.

Weights must be consecutive

Correct Incorrect Correct answer

Weights are powers of \(2\)

Correct Incorrect Correct answer

Weights \(\varphi^k\) form geometric and unique if no consecutive 1s

Correct Incorrect Correct answer

Only even digits allowed

Correct Incorrect Correct answer

297. Question

Prove count of \(n\)-digit numbers in base \(b\) is \((b – 1) \cdot b^{n-1}\).

Leading digit cannot be zero

Correct Incorrect Correct answer

Leading digit must be \(1\)

Correct Incorrect Correct answer

Digits must alternate

Correct Incorrect Correct answer

Only odd digits used

Correct Incorrect Correct answer

298. Question

Show for any base \(b > 1\), repeating 9s equal 1.

\(\frac{b-1}{b}\)

Correct Incorrect Correct answer

\(0\)

Correct Incorrect Correct answer

\(b – 1\)

Correct Incorrect Correct answer

\(1\)

Correct Incorrect Correct answer

299. Question

Convert \(0.124_{5}\) to base 10.

\(0.42\)

Correct Incorrect Correct answer

\(0.25\)

Correct Incorrect Correct answer

\(0.312\)

Correct Incorrect Correct answer

\(0.345\)

Correct Incorrect Correct answer

300. Question

Find integer whose base-6 is \(1324_6\) and base-7 is \(1133_7\).

Common integer is 340

Correct Incorrect Correct answer

Common integer is 416

Correct Incorrect Correct answer

Both represent same integer

Correct Incorrect Correct answer

No such integer exists

Correct Incorrect Correct answer

301. Question

Determine base where \(121_b\) is square of 11.

Works only for \(b = 10\)

Correct Incorrect Correct answer

Always a square

Correct Incorrect Correct answer

Only works for primes

Correct Incorrect Correct answer

Only valid for odd bases

Correct Incorrect Correct answer

302. Question

Show \(\sqrt{2}\) has infinite non-repeating expansion in every integer base.

\(\sqrt{2}\) is rational

Correct Incorrect Correct answer

\(\sqrt{2}\) can terminate in base 2

Correct Incorrect Correct answer

\(\sqrt{2}\) is irrational, hence non-repeating

Correct Incorrect Correct answer

Depends on base parity

Correct Incorrect Correct answer

303. Question

Convert \(\frac{1}{11}\) into base 11.

\(0.1_5\)

Correct Incorrect Correct answer

\(0.1_{11}\)

Correct Incorrect Correct answer

\(1.1_{11}\)

Correct Incorrect Correct answer

\(0.5_{11}\)

Correct Incorrect Correct answer

304. Question

Base-12 clock: 1 hour = \(100_{12}\) minutes; convert to decimal (sexagesimal).

1 h 60 min

Correct Incorrect Correct answer

2 h 24 min

Correct Incorrect Correct answer

2 h 12 min

Correct Incorrect Correct answer

3 h 0 min

Correct Incorrect Correct answer

305. Question

Show digit sum test for 11 in decimal uses alternating signs due to \(10 \equiv -1 \mod 11\).

\(N = \sum d_k \cdot 10^k\)

Correct Incorrect Correct answer

\(N = \sum d_k \cdot 11^k\)

Correct Incorrect Correct answer

\(N = \sum d_k \cdot k\)

Correct Incorrect Correct answer

\(N = \sum d_k \cdot (-1)^k\)

Correct Incorrect Correct answer

306. Question

Prove only bases that are powers of 10 use same digit symbols as decimal for same values.

Base must be binary

Correct Incorrect Correct answer

Base must have same digits as decimal

Correct Incorrect Correct answer

Positional weights must match powers of 10

Correct Incorrect Correct answer

Base must be 100

Correct Incorrect Correct answer

307. Question

Convert \(1111_2\) to base 3 efficiently.

\(120_3\)

Correct Incorrect Correct answer

\(111_3\)

Correct Incorrect Correct answer

\(121_3\)

Correct Incorrect Correct answer

\(112_3\)

Correct Incorrect Correct answer

308. Question

How many terminating fractions with denominator \(\leq 20\) in decimal?

32

Correct Incorrect Correct answer

40

Correct Incorrect Correct answer

47

Correct Incorrect Correct answer

20

Correct Incorrect Correct answer

309. Question

Convert \(0.\overline{02}_3\) to decimal.

\(0.2\)

Correct Incorrect Correct answer

\(0.33\)

Correct Incorrect Correct answer

\(0.5\)

Correct Incorrect Correct answer

\(0.25\)

Correct Incorrect Correct answer

310. Question

Show finite field \(\text{GF}(b)\) exists only if \(b\) is a prime power.

\(b\) must be even

Correct Incorrect Correct answer

\(b\) must be irrational

Correct Incorrect Correct answer

\(b\) must be a prime power

Correct Incorrect Correct answer

\(b\) must be less than 10

Correct Incorrect Correct answer

311. Question

Develop algorithm to express decimal integer directly in factorial number system.

Multiply by decreasing factorials

Correct Incorrect Correct answer

Repeated division by increasing divisors: 2, 3, 4…

Correct Incorrect Correct answer

Use modular exponentiation

Correct Incorrect Correct answer

Convert to binary first

Correct Incorrect Correct answer

312. Question

Show \(1/7\) repeats with period 3 in base 8.

Octal long division; observe remainder cycle length = 3

Correct Incorrect Correct answer

Binary conversion

Correct Incorrect Correct answer

Prime factor test

Correct Incorrect Correct answer

Convert to decimal first

Correct Incorrect Correct answer

313. Question

Prove optimal base for human cognition is approximately \(e \approx 2.718\).

It has shortest digits

Correct Incorrect Correct answer

It minimizes product of symbol count and length

Correct Incorrect Correct answer

It maximizes storage

Correct Incorrect Correct answer

It uses binary code

Correct Incorrect Correct answer

314. Question

Create base-phi numeral for 10.

\(10010_\varphi\)

Correct Incorrect Correct answer

\(10100_\varphi\)

Correct Incorrect Correct answer

\(10001_\varphi\)

Correct Incorrect Correct answer

\(11000_\varphi\)

Correct Incorrect Correct answer

315. Question

Show any integer is sum of distinct powers of 2 (binary uniqueness).

Binary division

Correct Incorrect Correct answer

Euclidean algorithm

Correct Incorrect Correct answer

Matrix exponentiation

Correct Incorrect Correct answer

Greedy subtraction algorithm

Correct Incorrect Correct answer

316. Question

Find smallest base where \(19_{10}\) is palindromic.

Base 16

Correct Incorrect Correct answer

Base 17

Correct Incorrect Correct answer

Base 18

Correct Incorrect Correct answer

Base 19

Correct Incorrect Correct answer

317. Question

Convert \(0.1_{12}\) to decimal.

\(0.0833\)

Correct Incorrect Correct answer

\(0.01\)

Correct Incorrect Correct answer

\(0.12\)

Correct Incorrect Correct answer

\(0.1\)

Correct Incorrect Correct answer

318. Question

Give general condition for \(n!\) ending in exactly \(k\) zeros in base \(b\).

\(n = b \cdot k\)

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\(k = \max \left( \frac{n}{b} \right)\)

Correct Incorrect Correct answer

\(b = p^k\)

Correct Incorrect Correct answer

\(\min_{p \mid b} \sum \left\lfloor \frac{n}{p^i} \right\rfloor \div v_p(b) = k\)

Correct Incorrect Correct answer

319. Question

Create bijection between base-\(b\) palindromes and binary strings of length \(\left\lfloor \frac{n}{2} \right\rfloor\).

Copy each digit from right to left

Correct Incorrect Correct answer

Fix trailing digits and rotate

Correct Incorrect Correct answer

Mirror second half from arbitrary first half

Correct Incorrect Correct answer

Use powers of \(b\) to form reflection

Correct Incorrect Correct answer

320. Question

Show repeating length of \(1/p\) in base \(b\) divides \(p – 1\).

Repeating length equals \(p + 1\)

Correct Incorrect Correct answer

Length divides \(b – 1\)

Correct Incorrect Correct answer

Always equal to \(p – 1\)

Correct Incorrect Correct answer

Order of \(b \mod p\) divides \(p – 1\)

Correct Incorrect Correct answer

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