Fermat's Little Theorem/Proof 3

Theorem
If $p$ is a prime number and $p \nmid n$, then $n^{p-1} \equiv 1 \pmod p$.

Proof
From the corollary to Reduced Residue System under Multiplication forms Abelian Group, the group of units of the ring $\Z / p \Z$ forms a group of order $p-1$ under multiplication.

Since $p \nmid n$, the residue class of $n$ is invertible modulo $p$ and thus an element of this group.

By Order of Element Divides Order of Finite Group, we have $n^{p-1} \equiv 1 \pmod p$.