Fermat's Little Theorem/Proof 4

Proof
Proof by induction over $n$.

Induction base:
 * $1^p \equiv 1 \pmod p$

Induction step:

Assume $n^p \equiv n \pmod p$

and so:

Dividing by $n$:
 * $\forall n: n^p \equiv n \pmod p \implies n^{p - 1} \equiv 1 \pmod p$