Fermat's Little Theorem/Corollary 1/Proof 1

Proof
There are two cases:


 * $(1): \quad$ If $p \divides n$, then $n^p \equiv 0 \equiv n \pmod p$.


 * $(2): \quad$ Otherwise, $p \nmid n$.

Then, by Fermat's Little Theorem, $n^{p-1} \equiv 1 \pmod p$.

Multiplying both sides by $n$, then by Congruence of Product we have:
 * $n^p \equiv n \pmod p$