How to show $((a,b),b)=(a,b)$? I want to prove $((a,b),b)=(a,b)$ for $a$, $b$ integers with some formal mathematical statement like Bezout's identity i.e., $(a,b)=d=am+bn$ or something else. 
Any ideas? 
Here $(a,b)$ means gcd.

My first sight is let $(a,b)=d$ and factors $a=d q_a$, $b=d q_b$ and proceed. 
I want more rigorous proof

Just note that $d=(a,b)$ is a divisor of $b$ and that $(x,y)=x$ whenever $x\mid y$.
