Call $gcd(a,b)=d$. Then $d|a$ and $d|b$. And if $c|a$ and $c|b$, then $c|d$.
It's simple to show that $d$ is SOME divisor of $a$ and $2a+b$, since we already know $d|a$ and $d|b$, so it divides the linear combination $2a+b$. However, how can I show that $d$ is the GREATEST common divisor of $a$ and $2a+b$?
Edit: Let $e$ be some other common divisor of $a$ and $2a+b$. Do we know if it's true that $e|b$? If so, we're finished since anything that divides both $a$ and $b$ will also divide $d$, since $d=gcd(a,b)$.