elementary number theory - $gcd(a,b) = gcd(a, a+2b)$ where $a$ is an odd integer

$a$ and $b$ are integers where $a$ is odd prove that $\gcd(a,b) = \gcd(a, a+2b)$

I know from $\gcd$ and divisibility of integer combinations that $\gcd(a,b)=d$
and that $d\mid a$ and $d\mid(a+2b)$, therefore $d$ is a common divisor of $a$ and $a+2b$. I'm having trouble with using the fact that $a$ is odd, and how to show that $d$ is the greatest common divisor. Thanks