If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even. Question:
If $ a^{2} + b^{2} = c^{2}$ and $c$ is even, prove that $a$ and $b$ are both even.
I am not quite sure how to prove this. My guess is proof by contradiction.
Assume the contrary, that is, $a^{2} + b^{2} = c^{2}$ for $c$ even and at least one of $a,b$ odd. 

Suppose $a^2+b^2=c^2$ and $c$ is even. Since  $a^2=c^2-b^2=(c+b)(c-b)$, $a$ and $b$ have the same parity. Assume $a$ is odd. Then $a^2+b^2\equiv 2(\mod4)$, whence $c^2\equiv2(\mod 4)$ which is a contradiction since $4\mid c^2$ as $2\mid c$. Hence $a$ and $b$ are even.
