Prove that if $a$, $b$, and $c$ are all positive integers such that $a^2$ + $b^2$ = $c^2$ then $a$ or $b$ must be even.
One way I was taught to handle implications with an or statement was to prove that if the first part is false, then the second part must be true. What I mean by this is that I can prove this implication to be correct if I can show that if $a$ is odd, then $b$ must be even. My problem is that I don't have much to work with. If $a$ is odd, then I could split the proof into 2 cases, one where $c$ is odd and the other where $c$ is even. I couldn't seem to get anywhere with that though.