modular arithmetic - Proof by Pairing Multiplicative Inverses

• Pairing up multiplicative inverses, show that $(p − 1)! \equiv −1 \pmod p$ for prime $p$


• Show that if $N$ is not prime, then $(N-1)! \not \equiv −1 \pmod N$. (Can use $d = \gcd(N, (N − 1)!)$ )

I know that all numberers in $(p-1)!$ can be grouped into $\frac{p-1}{2}$ paris that is congruent to 1 mod p. By I'm not sure how to go about proving it. Any suggestion is appreciated.

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