Prove that the greatest common factor of $m+n$ and $m^2+n^2$ is 1 or 2 if $m$ and $n$ are relatively prime. Prove that the greatest common factor of $m+n$ and $m^2+n^2$ is $1$ or $2$ if $m$ and $n$ are relatively prime natural numbers.
Can anyone give a step-by-step answer for this?

If prime $p$ divides both, $p$ divides $(m+n)^2-(m^2+n^2)=2mn$
If $p$ divides $m,p$ must divide $m+n-m=n$
$p$ must divide $(m,n)=1$
