<s>[INST] If $x^2+y^2=z^2$, why can't $x$ and $y$ both be odd? What does the following mean: If $x^2 + y^2 = z^2$ some integers $z$, then $x$ and $y$ can't be both odd (otherwise, the sum of their squares would be  $2$ modulo $4$, which can't be a square).  So, one of them must be even?
I see that if $x$ and $y$ are both odd, then $z^2 = 4k+2 =2(2k+1)$. So $z^2$ is even. But why does it say above that... can't be square?
 [/INST] If $n = 2k$, then $n^2 = 4k^2$ is a multiple of $4$.  
Likewise, if $n = 2k+1$, then $n^2 = 4(k^2 + k) + 1$.  
Therefore, a square is congruent to either $0$ or $1 \pmod{4}$.  In other words, a square is never of the form $4k + 2$, for some $k$.
More specifically, since you've seen that $x^2 + y^2 \equiv 2 \pmod{4}$ when $x$ and $y$ are odd, and since squares are never congruent to $2 \pmod{4}$, this shows that $x^2 + y^2$ is never a square when $x$ and $y$ are both odd.
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