If $a, b, c$ are integers with $a^2 + b^2 = c^2$, then $a$ and $b$ cannot both be odd If $a, b, c$ are integers with $a^2 + b^2 = c^2$, it's true that $a$ and $b$ cannot both be odd. 
But how can we prove it

The only possible squares $\mod 4$ are $1$ or $0$, corresponding to whether the original number is odd or even. If both $a$ and $b$ are odd, that means that $c^2 \equiv 1 + 1 \equiv 2\mod 4$, but this is impossible.
