Consider the equivalence relation $m\sim n$ defined to be $\frac {m-n}p=z$ (i.e., when $m-n$ is divisible by p) where $m,n,p,z\in \mathbb{Z}$ and $p$ is prime. Now suppose that $a$ is some integer not divisible by $p$. Prove that $[a]^{p-1}=[1]$ where [1] denotes the equivalence class of 1. Another common way to write this is $a^{p-1} \equiv 1$ $\ $(mod $p$).
I understand that there is some intent to prove multiplicative inverses here, but I'm a bit confused as to how one would approach it.