Fermat's Little Theorem/Proof 4

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

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:

And so $\forall n: n^p \equiv n \pmod p \implies n^{p-1} \equiv 1 \pmod p$ (Dividing by $n$).