Fermat, Wilson and Lagrange Theorems - One Line Questions
1.
If n is a composite number, what can be said about (n-1)! mod n? —
(n-1)! = 0 (mod n)
2.
Wilson's Theorem states that a positive integer n is a prime number if and only if which condition holds? —
(n-1)! = -1 (mod n)
3.
If n is a composite number, which of the following is generally true for (n-1)! mod n? —
(n-1)! = 0 (mod n) for n > 4
4.
If H is a subgroup of a finite group G, then the number of distinct left cosets of H in G is equal to: —
|G|/|H|
5.
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? —
|H| divides |G|
6.
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? —
1
7.
If p is a prime number, what is the value of (p-1)! mod p? —
p-1
8.
What is the value of 3^5 mod 5 according to Fermat's Little Theorem? —
3
9.
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? —
1
10.
What is the value of (7-1)! mod 7? —
6
11.
What is the value of 4^3 mod 3? —
1
12.
What is the value of (5-1)! mod 5? —
4
13.
What is the value of 10! mod 11? —
10
14.
What is the value of 2^7 mod 7? —
2
15.
What is the value of (p-1)! mod p for a prime p, according to Wilson's Theorem? —
p-1
16.
If n is a composite number, and n = ab where 1 < a, b < n and a != b, then (n-1)! mod n is: —
0
17.
If n is a composite number and n > 4, then (n-1)! is congruent to what modulo n? —
0
18.
Fermat's Little Theorem states that for a prime p and integer a, a^p is congruent to what modulo p? —
a
19.
Consider the group of integers modulo 6 under addition, Z_6. What is the order of the subgroup generated by 3? —
2
20.
Consider the group S_3 (the symmetric group on 3 elements). What is its order? —
6
21.
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? —
5
22.
Consider the group of non-zero integers modulo 7 under multiplication, (Z/7Z)*. What is the order of this group? —
6
23.
Fermat's Little Theorem is often used in primality testing. What is a 'Fermat pseudoprime'? —
A composite number n that satisfies a^(n-1) = 1 (mod n) for some base a.
24.
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 must not be divisible by p
25.
Fermat's Little Theorem is a special case of Euler's totient theorem when the modulus is: —
A prime number
26.
The condition a^(n-1) = 1 (mod n) for all integers 'a' such that gcd(a, n) = 1 is known as the: —
Fermat pseudoprime condition
27.
Which theorem provides a criterion for primality testing based on factorials? —
Wilson's Theorem
28.
Which theorem states that for any finite group G, the order of any element g in G divides the order of G? —
Lagrange's Theorem
29.
Which theorem is primarily concerned with the structure of finite groups and the divisibility of their orders? —
Lagrange's Theorem
30.
If H is a subgroup of S_3 with order 2, what does Lagrange's Theorem guarantee? —
The order of H must divide 6.
31.
What is the converse of Fermat's Little Theorem? —
If a^n = a (mod n) for all a, then n is prime.
32.
What is the direct implication of Fermat's Little Theorem for modular arithmetic? —
It simplifies the calculation of powers modulo a prime.
33.
Fermat's Little Theorem is particularly useful in cryptography for operations involving: —
Modular exponentiation with prime moduli
34.
Lagrange's theorem implies that if a group G has order n, then the order of any element g in G must be a: —
Divisor of n
35.
Consider the group of integers modulo n under addition, Z_n. What is the order of this group? —
n
36.
If n is a composite number greater than 4, it is guaranteed that (n-1)! is divisible by n if: —
n is a square of a prime
37.
If G is a finite group and g is an element of G, the set {g^k | k is an integer} forms a: —
Cyclic subgroup
38.
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: —
Index of H in G
39.
Fermat's Little Theorem can be generalized by Euler's totient theorem. What is the exponent in Euler's theorem? —
phi(n)
40.
Wilson's Theorem is primarily a test for: —
Prime numbers
41.
The number of distinct subgroups of a cyclic group of order n is equal to the number of: —
Divisors of n
42.
Lagrange's Theorem is fundamental in the study of which mathematical structure? —
Groups
43.
In the context of Lagrange's Theorem, what is meant by the 'order' of a group? —
The number of elements in the group.
44.
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? —
The order of a subgroup divides the order of the group.
45.
Which of the following is a direct consequence of Wilson's Theorem? —
The primality of numbers like 3, 5, 7.
46.
What is the relationship between Fermat's Little Theorem and Wilson's Theorem? —
Fermat's Little Theorem implies Wilson's Theorem.
47.
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? —
Fermat's Little Theorem
48.
The statement 'If p is a prime number, then a^p = a (mod p) for any integer a' is a form of: —
Fermat's Little Theorem
49.
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? —
Fermat's Little Theorem
50.
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? —
No, not always