 Solution

If $p$ is a prime number and $p\nmid a$, multiplication by $a$ permutes the nonzero residue classes modulo $p$: $ai\equiv aj$ implies $i\equiv j$ because $a$ has a multiplicative inverse. Multiplying all the permuted classes givesThe factorial is a nonzero element of the finite field ${\mathbb{F}}_p$, so cancellation is valid. Thereforewhich is Fermat's little theorem.