Question

In number theory, Wilson’s theorem states that a natural number n > 1 is prime
if and only if (n − 1)! ≡ −1 (mod n).
(a) Check that 5 is a prime number using Wilson’s theorem.
(b) Let n and m be natural numbers such that m divides n. Prove the following statement

“For any integer a, if a ≡ −1 (mod n), then a ≡ −1 (mod m).”

You may need this fact in doing (c).
(c) The “if” part of Wilson’s theorem says that

“A natural number n > 1 is prime if (n − 1)! ≡ −1 (mod n).”

Prove this statement by contradiction.

Answer 1