how to prove that a natural number n > 1 is prime if and only if n divides (n-2)! - 1?

Question

how to prove that a natural number n > 1 is prime if and only if n divides (n-2)! - 1? I know it is a 'iff' questions so that it need to be proved by both directions, and I tried to prove by contradiction or contrapositive but still did not figure it out.

Answer 1

Wilson's theorem states that a natural number $n $ is a prime number if and only if $$(n-1)! \equiv -1 \pmod n$$

$$(n-1)(n-2)! \equiv -1 \pmod n$$

$$(n-2)! \equiv 1 \pmod n$$

Answer 2

Wilson's Theorem:

$n$ is prime $\Leftrightarrow (n-1)!+1\equiv 0(\mod n)$

But as $(n-1)!+1\equiv(n-1)((n-2)!-1)(\mod n)$ and gcd$(n,n-1)=1$, then

$(n-2)!-1\equiv0(\mod n) \Leftrightarrow (n-1)!+1\equiv0(\mod n) \Leftrightarrow n$ is prime