Fermat, Wilson and Lagrange Theorems - Question Bank

1. 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
2. Wilson's Theorem is primarily a test for:
A) Perfect numbers
B) Prime numbers
C) Composite numbers
D) Abundant numbers
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. What is the value of 2^7 mod 7?
A) 0
B) 1
C) 2
D) 4
11. 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.
12. 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
13. 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
14. What is the value of 10! mod 11?
A) 0
B) 1
C) 10
D) 11
15. 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.
16. 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
17. 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
18. What is the value of (5-1)! mod 5?
A) 0
B) 1
C) 2
D) 4
19. 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
20. 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
21. 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)
22. What is the value of 4^3 mod 3?
A) 0
B) 1
C) 2
D) 4
23. 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
24. 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|
25. 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
26. What is the value of (7-1)! mod 7?
A) 0
B) 1
C) 6
D) 7
27. 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
28. 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
29. 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
30. 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
31. What is the value of 3^5 mod 5 according to Fermat's Little Theorem?
A) 0
B) 1
C) 3
D) 4
32. 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
33. 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.
34. Consider the group S_3 (the symmetric group on 3 elements). What is its order?
A) 2
B) 3
C) 6
D) 9
35. 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.
36. 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
37. 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.
38. Lagrange's Theorem is fundamental in the study of which mathematical structure?
A) Rings
B) Fields
C) Groups
D) Vector Spaces
39. 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
40. 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
41. 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.
42. 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
43. 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).
44. 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
45. 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
46. 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.
47. If p is a prime number, what is the value of (p-1)! mod p?
A) 0
B) 1
C) p-1
D) p
48. 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)
49. 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
50. 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
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