I'm studying for a mathematical induction test tomorrow, and I have a practice question:
Use mathematical induction to prove that if $a$ and $b$ are integers with $a \equiv b \pmod m$ then $a^k \equiv b^k \pmod m$ for all $k \ge 0$
So I've tried tackling the base case $P(0)$, since $k$ can equal $0$.
$i=k$
Assume $P(i)$ is true.
Prove $P(i+1)$
$a^{i+1}\equiv b^{i+1} \pmod m$
Base Case: $P(0)$
$a^{i+0}\equiv b^{i+0} \pmod m$
$a^i\equiv b^i \pmod m$ is equal to our original equation.
So $P(0)$ is true.
How can this possibly make sense? I don't understand how we can come to a logical conclusion with an assumption of proof. I do not see how this proves the base case.
Could someone show me how to tackle the inductive step? I'm not even sure where to start once I change $k$ to $i+1$