<s>[INST] Suppose $a,b$ are integers. If $4\mid(a^2 +b^2)$, then $a$ and $b$ are not both odd. I've tried using direct proof and contrapositive proof to prove this.
I'm stuck. Is anyone able to help solve or give me some hints?
 [/INST] If $4|a^2+b^2$ then $a^2+b^2=4m$ for some integer $m$.
Now suppose that $a$ and $b$ are both odd. Then $a=2i+1$ and $b=2j+1$ for some integers $i,j$.
Then $a^2+b^2=4i^2+4i+1+4j^2+4j+1=4(i^2+i+j^2+j)+2$, which is a contradiction.
If you are comfortable with modular arithmetic, we would say that we can't have both $a^2+b^2\equiv 0$ mod $4$ AND $a^2+b^2\equiv 2$ mod $4$.
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