Number theory: prove that if $a,b,c$ odd then $2\gcd(a,b,c) = \gcd(a+b,b+c, c+a)$ Please help! Am lost with the following:
Prove that if $a,b,c$ are odd integers, then $2 \gcd(a,b,c) = \gcd( a+b, b+c, c+a)$
Thanks a lot!!

Hint $\,(a,b,c)\to (a\!+\!b,b\!+\!c,c\!+\!a)$ is linear with det $= \color{#c00}2\,$ so by the simple method in this answer 
$$ \gcd(a,b,c)\mid  \gcd(a\!+\!b,b\!+\!c,c\!+\!a)\mid \color{#c00} 2 \gcd(a,b,c)$$
