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

We can prove this by contradiction.
$\text{Let }\;\; a = 2m+1, b = 2n + 1 $
So, the LHS is:
$$
\begin{align}
a^2 + b^2 &= 4(m^2+n^2) + 4(m+n) + 2 \\\\ 
          &= 2\left( 2(m^2+n^2) + 2(m+n)+1 \right) \\\\
          &= 2\times\text{an odd number}
\end{align}
$$
This can not be a perfect square, as it doesn't have an even power of $2$.
