8.02k-o Euclid's lemma, Fermat's little theorem and the binomial theorem Quiz: OCR Further Maths, Unit 5

The 20 questions

1. According to Euclid's lemma, if a prime p divides ab, what must p divide?

   • a or b
   • a and b
   • a - b
   • ab + 1

2. In Fermat's Little Theorem, what condition must prime p and integer a satisfy?

   • coprime
   • even
   • equal
   • consecutive

3. What is the value of 2^6 modulo 7 according to Fermat's Little Theorem?

   • 1
   • 2
   • 0
   • 6

4. Using Fermat's Little Theorem, what is the value of 3^10 modulo 11?

   • 10
   • 3
   • 1
   • 0

5. By Fermat's Little Theorem, what is the modular inverse of a modulo prime p?

   • a^p mod p
   • a^(1-p) mod p
   • a^(p-1) mod p
   • a^(p-2) mod p

6. Using Fermat's Little Theorem, what is the multiplicative inverse of 2 modulo 7?

   • 1
   • 4
   • 3
   • 5

7. What is the coefficient of x^k in the binomial expansion of (1 + x)^n?

   • k choose n
   • n factorial
   • n squared
   • n choose k

8. What is the remainder when 5^12 is divided by 13?

   • 1
   • 5
   • 0
   • 12

9. According to Fermat's Little Theorem, what is 4^7 modulo 7?

   • 0
   • 6
   • 1
   • 4

10. What is the remainder when 3^31 is divided by 7?

    • 1
    • 5
    • 3
    • 6

11. For any integer a and prime p, what is a^p congruent to modulo p?

    • 1
    • p - 1
    • a
    • 0

12. What is the sum of all binomial coefficients for a given positive integer n?

    • n!
    • 2^n
    • 2n
    • n^2

13. For a prime p, which integer divides all binomial coefficients p choose k for 0 < k < p?

    • 2
    • p + 1
    • p
    • p - 1

14. Euclid's lemma is key to proving which fundamental theorem of arithmetic?

    • Binomial expansion
    • Fermat's last
    • Unique factorisation
    • Prime number

15. What is the remainder when 2^100 is divided by 11?

    • 1
    • 2
    • 10
    • 4

16. What is the sum of alternating binomial coefficients for any integer n > 0?

    • 0
    • -1
    • 2^n
    • 1

17. If x is coprime to prime p and a ≡ b mod (p - 1), what is x^a mod p?

    • a^b
    • x^(p-1)
    • 1
    • x^b

18. What is the remainder when 7^82 is divided by 13?

    • 4
    • 9
    • 7
    • 1

19. In each term of the binomial expansion of (a + b)^n, what is the sum of exponents?

    • n
    • 2n
    • n + 1
    • n - 1

20. How many terms are in the expanded form of (a + b)^n for a positive integer n?

    • n
    • n + 1
    • n - 1
    • 2^n

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