Congruence Implication with $2^k$

Question

For $k \geq 3$ I have to show the following implication: $$ (-1)^i5^j \equiv (-1)^m5^n \mod2^k \Rightarrow (-1)^i \equiv (-1)^m \mod4$$

I have tried a few things but I couldn´t solve it, I would appreciate any help!

Answer 1

For $k\ge 3\;$ (and even $k\ge 2$), \begin{align} (-1)^i5^j \equiv (-1)^m5^n \bmod2^k &\implies (-1)^i5^j \equiv (-1)^m5^n \bmod2^2=4 \\ &\iff (-1)^i 1^j\equiv(-1)^m 1^n\quad\text{ since } 5\equiv 1 \bmod 4 \\ &\iff (-1)^i \equiv(-1)^m \end{align}