<s>[INST] 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. 
 [/INST] $c$ is even, thus $c^2=4n$ for some integer $n$.
Now if $a,b$ are both odd, then 
$$a^2+b^2=(2k+1)^2 + (2m +1)^2$$
$$=4k^2+ 4k + 1 + 4m^2 + 4m+ 1$$
$$=4n' + 2 \equiv 2 \pmod4$$
Thus $4 \nmid (a^2+b^2)$, while $4 \mid c^2$ - contradiction.
Also it is obvious that if $a$ even and $b$ odd, or $a$ odd and $b$ even; then $a^2+b^2$ is odd, while $c^2$ is even - contradiction.
Thus it has to be that both $a$, $b$ are even.
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