8.02e-i Finite (modular) arithmetic and prime numbers Quiz: OCR Further Maths, Unit 5

The 20 questions

1. Which condition is equivalent to the congruence statement a ≡ b (mod m)?

   • b divides a + m
   • m divides a - b
   • m divides a + b
   • a divides b - m

2. For a prime p and integer a coprime to p, what is a^(p-1) (mod p)?

   • p - 1
   • a
   • 1
   • 0

3. What is the value of Euler's totient function phi(p) for any prime p?

   • p
   • p - 1
   • 1
   • p + 1

4. Using Fermat's Little Theorem, what is the value of 7^10 (mod 11)?

   • 4
   • 7
   • 10
   • 1

5. What is the multiplicative inverse of 3 modulo 7?

   • 4
   • 3
   • 5
   • 2

6. What is the value of Euler's totient function phi(12)?

   • 6
   • 4
   • 8
   • 2

7. When does the linear congruence ax ≡ b (mod m) have a unique solution?

   • gcd(b, m) = 1
   • a > m
   • gcd(a, b) = 1
   • gcd(a, m) = 1

8. What is the smallest positive integer solution to 2x ≡ 1 (mod 5)?

   • 2
   • 1
   • 3
   • 4

9. According to Wilson's Theorem, what is (p - 1)! (mod p) for prime p?

   • 1
   • 0
   • p / 2
   • p - 1

10. Using Wilson's Theorem, what is the value of 4! (mod 5)?

    • 1
    • 0
    • 2
    • 4

11. The Chinese Remainder Theorem requires the moduli m1 and m2 to be what?

    • Equal
    • Consecutive integers
    • Both prime
    • Pairwise coprime

12. Using Fermat's Little Theorem, what is the value of 3^20 (mod 7)?

    • 1
    • 2
    • 3
    • 4

13. What is the value of Euler's totient function phi(15)?

    • 14
    • 2
    • 8
    • 4

14. What is the smallest positive integer solving x ≡ 2 (mod 3) and x ≡ 3 (mod 5)?

    • 5
    • 11
    • 13
    • 8

15. For coprime integers a and m, what is a^phi(m) (mod m) equal to?

    • 1
    • phi(m)
    • 0
    • a

16. How many incongruent solutions modulo 6 does 2x ≡ 4 (mod 6) have?

    • 2
    • 1
    • 0
    • 3

17. What multiplicative order must an element modulo p have to be a primitive root?

    • p - 1
    • 1
    • p
    • (p - 1)/2

18. What is the multiplicative order of 2 modulo 5?

    • 2
    • 4
    • 5
    • 3

19. What is the remainder when 2^100 is divided by 13?

    • 3
    • 9
    • 1
    • 4

20. For prime p and positive integer k, what is the value of phi(p^k)?

    • p^k - p^(k-1)
    • p^k - 1
    • (p - 1)^k
    • p^(k-1)

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