TRB BT Assistant - Fermat, Wilson and Lagrange Theorems Online Test (MCQ Quiz)

1. Which theorem states that if p is a prime number, then for any integer a, the number a^p - a is an integer multiple of p?

A) Wilson's Theorem B) Lagrange's Theorem C) Fermat's Little Theorem D) Chinese Remainder Theorem

2. According to Fermat's Little Theorem, if p is a prime and a is an integer not divisible by p, what is the value of a^(p-1) modulo p?

A) 0 B) 1 C) p D) p-1

3. Wilson's Theorem states that a positive integer n is a prime number if and only if which condition holds?

A) (n-1)! = 1 (mod n) B) (n-1)! = 0 (mod n) C) (n-1)! = -1 (mod n) D) (n-1)! = n (mod n)

4. If p is a prime number, what is the value of (p-1)! mod p?

A) 0 B) 1 C) p-1 D) p

5. Lagrange's Theorem, in the context of group theory, relates the order of a subgroup to the order of the group. What is the statement of Lagrange's Theorem?

A) The order of a subgroup divides the order of the group. B) The order of an element divides the order of the group. C) The order of the group is the product of the orders of its subgroups. D) The order of a normal subgroup divides the order of the quotient group.

6. Consider the group of integers modulo n under addition, Z_n. What is the order of this group?

A) n B) n-1 C) 1 D) 0

7. If G is a finite group and H is a subgroup of G, which of the following must be true according to Lagrange's Theorem?

A) |H| = |G| B) |G| divides |H| C) |H| divides |G| D) |H| = 1

8. What is the direct implication of Fermat's Little Theorem for modular arithmetic?

A) It simplifies the calculation of powers modulo a prime. B) It guarantees that all integers are congruent modulo p. C) It proves that p is always divisible by a. D) It states that a^p is always congruent to a (mod p^2).

9. Which theorem provides a criterion for primality testing based on factorials?

A) Fermat's Little Theorem B) Lagrange's Theorem C) Wilson's Theorem D) Euler's Theorem

10. What is the converse of Fermat's Little Theorem?

A) If a^(n-1) = 1 (mod n) for all a coprime to n, then n is prime. B) If a^n = a (mod n) for all a, then n is prime. C) If (n-1)! = -1 (mod n), then n is prime. D) If n is prime, then a^(n-1) = 1 (mod n) for all a.

11. The condition a^(n-1) = 1 (mod n) for all integers 'a' such that gcd(a, n) = 1 is known as the:

A) Carmichael condition B) Fermat pseudoprime condition C) Wilson's condition D) Lagrange condition

12. If n is a composite number, what can be said about (n-1)! mod n?

A) (n-1)! = 0 (mod n) B) (n-1)! = 1 (mod n) C) (n-1)! = -1 (mod n) D) (n-1)! can be anything modulo n

13. Lagrange's Theorem is fundamental in the study of which mathematical structure?

A) Rings B) Fields C) Groups D) Vector Spaces

14. In the context of Lagrange's Theorem, what is meant by the 'order' of a group?

A) The number of generators of the group. B) The number of elements in the group. C) The smallest positive integer k such that g^k = e for all g in the group. D) The exponent of the group.

15. Fermat's Little Theorem can be generalized by Euler's totient theorem. What is the exponent in Euler's theorem?

A) p-1 B) phi(n) C) n D) a

16. Which of the following is a direct consequence of Wilson's Theorem?

A) The primality of numbers like 3, 5, 7. B) The existence of infinite primes. C) The fundamental theorem of arithmetic. D) The structure of cyclic groups.

17. Consider the group S_3 (the symmetric group on 3 elements). What is its order?

A) 2 B) 3 C) 6 D) 9

18. If H is a subgroup of S_3 with order 2, what does Lagrange's Theorem guarantee?

A) H must be the trivial subgroup. B) The order of H must divide 6. C) H must be isomorphic to S_3. D) The order of S_3 must divide the order of H.

19. Fermat's Little Theorem is particularly useful in cryptography for operations involving:

A) Large composite numbers B) Modular exponentiation with prime moduli C) Factorization D) Discrete logarithms in non-prime moduli

20. What is the value of 3^5 mod 5 according to Fermat's Little Theorem?

A) 0 B) 1 C) 3 D) 4

21. If n is a composite number greater than 4, it is guaranteed that (n-1)! is divisible by n if:

A) n is a square of a prime B) n is a product of two distinct primes C) n is any composite number D) n is an even composite number

22. Which theorem states that for any finite group G, the order of any element g in G divides the order of G?

A) Fermat's Little Theorem B) Wilson's Theorem C) Lagrange's Theorem D) Cauchy's Theorem

23. If G is a cyclic group of order n, and d is a divisor of n, how many subgroups of order d does G have?

A) 0 B) 1 C) d D) n/d

24. The statement 'If p is a prime number, then a^p = a (mod p) for any integer a' is a form of:

A) Wilson's Theorem B) Lagrange's Theorem C) Fermat's Little Theorem D) Euler's Theorem

25. What is the value of (7-1)! mod 7?

A) 0 B) 1 C) 6 D) 7

26. Consider the group of non-zero integers modulo 7 under multiplication, (Z/7Z)*. What is the order of this group?

A) 6 B) 7 C) 5 D) 1

27. If H is a subgroup of a finite group G, then the number of distinct left cosets of H in G is equal to:

A) |H| B) |G| C) |G|/|H| D) |H|/|G|

28. Fermat's Little Theorem is a special case of Euler's totient theorem when the modulus is:

A) Any integer B) A power of a prime C) A prime number D) A perfect square

29. What is the value of 4^3 mod 3?

A) 0 B) 1 C) 2 D) 4

30. If n is a composite number, which of the following is generally true for (n-1)! mod n?

A) (n-1)! = 1 (mod n) B) (n-1)! = -1 (mod n) C) (n-1)! = 0 (mod n) for n > 4 D) (n-1)! = n-1 (mod n)

31. Lagrange's theorem implies that if a group G has order n, then the order of any element g in G must be a:

A) Multiple of n B) Prime factor of n C) Divisor of n D) Power of n

32. Which theorem is used to prove that if p is a prime, then the polynomial x^p - x has p roots in the field Z_p?

A) Wilson's Theorem B) Fermat's Little Theorem C) Lagrange's Theorem D) Gauss's Lemma

33. What is the value of (5-1)! mod 5?

A) 0 B) 1 C) 2 D) 4

34. If G is a finite group and g is an element of G, the set {g^k | k is an integer} forms a:

A) Normal subgroup B) Coset C) Cyclic subgroup D) Trivial subgroup

35. Lagrange's Theorem provides a necessary condition for a number to be the order of an element in a finite group. Is it also a sufficient condition?

A) Yes, always B) No, not always C) Yes, if the group is cyclic D) Yes, if the group is abelian

36. Fermat's Little Theorem is often used in primality testing. What is a 'Fermat pseudoprime'?

A) A composite number n that satisfies a^(n-1) = 1 (mod n) for some base a. B) A prime number n that satisfies a^(n-1) = 1 (mod n) for some base a. C) A composite number n that satisfies a^n = a (mod n) for all bases a. D) A prime number n that satisfies a^n = a (mod n) for all bases a.

37. What is the value of 10! mod 11?

A) 0 B) 1 C) 10 D) 11

38. If n is a composite number, and n = ab where 1 < a, b < n and a != b, then (n-1)! mod n is:

A) 1 B) -1 C) 0 D) a

39. The number of distinct subgroups of a cyclic group of order n is equal to the number of:

A) Prime factors of n B) Elements in the group C) Divisors of n D) Generators of the group

40. What is the relationship between Fermat's Little Theorem and Wilson's Theorem?

A) They are equivalent statements. B) Fermat's Little Theorem implies Wilson's Theorem. C) Wilson's Theorem implies Fermat's Little Theorem. D) They are unrelated.

41. What is the value of 2^7 mod 7?

A) 0 B) 1 C) 2 D) 4

42. If G is a finite group and H is a subgroup, the set of left cosets of H in G forms a partition of G. The number of these cosets is called the:

A) Order of H B) Index of H in G C) Order of G D) Lagrangian number

43. Which theorem is primarily concerned with the structure of finite groups and the divisibility of their orders?

A) Fermat's Little Theorem B) Wilson's Theorem C) Lagrange's Theorem D) Chinese Remainder Theorem

44. What is the value of (p-1)! mod p for a prime p, according to Wilson's Theorem?

A) 0 B) 1 C) p-1 D) p

45. If n is a composite number and n > 4, then (n-1)! is congruent to what modulo n?

A) 1 B) -1 C) 0 D) n-1

46. Fermat's Little Theorem states that for a prime p and integer a, a^p is congruent to what modulo p?

A) 1 B) a C) p D) 0

47. Consider the group of integers modulo 6 under addition, Z_6. What is the order of the subgroup generated by 3?

A) 1 B) 2 C) 3 D) 6

48. If G is a group of order 12, which of the following cannot be the order of a subgroup of G, according to Lagrange's Theorem?

A) 2 B) 3 C) 5 D) 6

49. Wilson's Theorem is primarily a test for:

A) Perfect numbers B) Prime numbers C) Composite numbers D) Abundant numbers

50. Fermat's Little Theorem states that if p is a prime number, then for any integer a, a^p ≡ a (mod p). What is the condition on 'a' for the alternative form a^(p-1) ≡ 1 (mod p)?

A) a must be divisible by p B) a must not be divisible by p C) a must be positive D) a must be odd

Fermat, Wilson and Lagrange Theorems - Online Test

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