If $p$ is a prime, then $(p-1)! \equiv -1 (mod p)$. Hint: $(p-1)!$ is the product of elements in $Z_p$. Match each element to its inverse.
I can understand by testing some primes that for any prime $(p-1)!+1$ is a multiple of that prime, but I'm not sure how to prove it in general or how inverses play into it.