The base case is pretty straightforward. But I'm stuck on the inductive step.
As the base case holds, assume for when $n=k$ holds, show the $k+1$ case holds true.
Inductive Hypothesis: $a^k \equiv b^k \pmod m$, then
$$a^{k+1} \equiv b^{k+1} \pmod m \iff a^{k+1} - b^{k+1} = m(k), k \in \mathbb{Z}. $$
I think I'm missing some steps, I'm not sure how to manipulate what I have to shows it holds.
It follows from the base case and induction hypothesis: $a^{k+1}=a.a^k\equiv b.b^k=b^{k+1}.$