In what case we get this relation: $a^{m}≡b^{m}(mod(c))$

Question

Let $a,b$ and $c$ three natural numbers such that $a≡b \pmod{c}$.

I am asking when getting relation $a^{m}≡b^{m}\pmod{c}$, in which $m \in \mathbb{N}$

Answer 1

For any $m \in \mathbb{Z}^+$, and any $a,b \in \mathbb{R}$, we have $$a^m-b^m = (a-b)(a^{m-1}+a^{m-2}b + \cdots + a b^{m-2} + b^{m-1})$$ Now conclude what you want.