<s>[INST] 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?
 [/INST] Let $d\lvert m+n$ and $d\lvert m^2+n^2$.
Then $d\lvert(m^2+n^2)-(m-n)(m+n)$, so $d\lvert 2n^2$;
$\;$and $d\lvert(m^2+n^2)+(m-n)(m+n)$, so $d\lvert 2m^2$.
Since $\gcd(2m^2, 2n^2)=2\gcd(m^2,n^2)=2\cdot1=2$, 
$\hspace{.3 in}d\lvert2$ so $d=1$ or $d=2$.
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