1. ## Congruences

Using the fact that $aa' \equiv bb' (mod m)$, prove that if $a \equiv b (mod m)$ then $a^e \equiv b^e (mod m)$ for any $e \geq 0$.

2. Originally Posted by Zennie
Using the fact that $aa' \equiv bb' (mod m)$, prove that if $a \equiv b (mod m)$ then $a^e \equiv b^e (mod m)$ for any $e \geq 0$.
are you sure u hv the complete question here?

3. Originally Posted by Zennie
Using the fact that if $\color{red}a\equiv b\!\pmod m$ and $\color{red}a'\equiv b'\!\pmod m$ then $\color{red}aa' \equiv bb'\!\pmod m\color{black},$ prove that if $a \equiv b (mod m)$ then $a^e \equiv b^e (mod m)$ for any $e \geq 0$.
Use induction on $e.$