Fermat's Little Theorem/Corollary 1/Proof 1

Corollary to Fermat's Little Theorem
If $p$ is a prime number, then $n^p \equiv n \pmod p$.

Proof
There are two cases:


 * $(1):\quad$ If $p \mathop \backslash n$, 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 $n^p \equiv n \pmod p$.