I was trying to do this proof where:
Assume $a,b,r,s$ are relatively prime, and that $$a^2+b^2=r^2$$ and $$a^2-b^2=s^2$$ Prove that $a,r,s$ are odd and $b$ is even.
So I started off by saying that if a number $x^2$ is odd, then $x$ is odd. Same applies for even numbers. Then I said that both $a$ and $b$ can't be even at the same time, and can't be odd, since if they are both odd, $r$ and $s$ will both be even. But then $b$ can be odd while $a$ is even and vice versa: both $r$ and $s$ will be odd. Am I missing something?