I need to prove that
$(ah,bk)=(a,b)(h,k)\left( \frac{a}{(a,b)},\frac{k}{(h,k)}\right)\left( \frac{b}{(a,b)},\frac{h}{(h,k)}\right)$
I'm most certain I need to use $\left(\frac{a}{(a,b)},\frac{b}{(a,b)}\right)=1 $ and $(a ,b)=1 $ and $a \mid bc $ then $a \mid c$ but I'm not sure how to build up the proof.