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

Hint $\,\ ((a,b),b) = (a,(b,b)) = (a,b)\ $ by the gcd Associative Law.

Or directly $\,\ d\mid((a,b),b)\iff d\mid(a,b),b\iff d\mid a,b,\, b\iff d\mid a,b\iff d\mid (a,b)\,$ where we used the gcd Universal Property $3$ times.

Or $\ d\mid b\,\Rightarrow\, (d,b) = (d, b\bmod d) = (d,0) = d\,$ by the Euclidean algorithm, or  $\,(d,b) = d(1,b/d) = d\,$ by the gcd Distributive Law.

Or use $\ (a\Bbb Z + b\Bbb Z) + b\Bbb Z = a\Bbb Z + (b\Bbb Z + b\Bbb Z) = a\Bbb Z + b\Bbb Z\ $ and Bezout.
