<s>[INST] Show that $\gcd(a + b, a^2 + b^2) = 1$ or $2$ if $\gcd(a, b)=1$ Show that $\gcd(a + b, a^2 + b^2) = 1$ or $2$ if $\gcd(a, b)=1$.
I have absolutely no clue where to start and what to do, please provide complete proof and answer.
 [/INST] Suppose that $\gcd(a+b,a^2+b^2)=d$
Then $d \mid a+b$ and $d \mid a^2+b^2$.
$d \mid (a+b)(a+b)-(a-b)(a+b)=2a^2$.
$d \mid (a+b)(a+b)+(a-b)(a+b)=2b^2$
$d \mid (2a^2,2b^2) = 2(a^2,b^2)=2(a,b)^2=2.1=2 \implies d=1$ or $2$.
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