If $a^n$ is equivalent to $b^n$ (mod c) then does this imply a is equivalent to b (mod c)?

Question

I see the property that a equivalent to b (mod c) implies a^n is equivalent to b^n (mod c) for a non-negative integer n, but cannot figure out if it works in the other direction. A link to the property or proof, or a counterexample would be appreciated.

I am not new to math but am new to number theory. Thanks for your time.

Answer 1

No. For example $2^4\equiv 1^4 \pmod 5$,but $2\not\equiv 1 \pmod 5$.

Answer 2

We have $$1^2 \equiv (-1)^2 \pmod{ c}$$

That is $$1^2 \equiv (c-1)^2 \pmod{ c}$$

However $1 \not \equiv c-1 \pmod{c} $ for any $c\ge3$.