One number-theoretic result mentioned in the course was Wilson’s Theorem : If p is a prime then (p − 1)! is congruent to −1 mod p . — The purpose of this exercise is to show the reverse implication....

One number-theoretic result mentioned in the course was
Wilson’s


Theorem:
If p is a prime then (p − 1)! is congruent to −1 mod p. — The purpose of this exercise is to show the reverse implication.

(a) Suppose n > 1 is a composite integer ab where a and b are
unequal
integers both greater than 1. Prove that (n − 1)! is congruent to 0 mod n. [Hint: Why are both factors less than n/2?]

(b) The preceding part of the problem proves the reverse implication unless n = p²
where p is a prime. Prove that if p > 2 is prime then (p²− 1)! is congruent to 0 mod p², and find k ∈ {0, 1, 2, 3} such that (2²
− 1)! is congruent to k mod 4.

Please answer part b of this question.