<s>[INST] Prove that if $x$ is odd then $x^2 + (x + 2)^2$ is divisible by $2$ but not for $4$ This is what I tried:
$$x^2 + x^2 + 4x + 4=2x^2 + 4(x + 1),$$
so it's divisible by $2$, since this expression is a sum of a multiple of $2$ and a multiple of $4$.
Therefore, for the expression not to be a multiple of $4$, $2x^2$ can't be multiple of $4$, that's what I can't prove.
Brazilian student, sorry for my English.
 [/INST] Perhaps the easiest way is to put $x=2k-1$, $k$ an integer, since $x$ is odd. Then
$$ x^2+(x+2)^2 = (2k-1)^2+(2k+1)^2 = 8k^2+2, $$
which is clearly $2$ more than a multiple of $4$, and hence not a multiple of $4$ (but is $2(4k^2+1)$, so obviously a multiple of $2$).
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