This isn't a duplicate! the "duplicate" question doesn't use my method of proof but other methods and I made it clear that I want this method in specific.
I want to prove the following claim using the following method.
$(ax,bx)=x(a,b)$
What I have done so far:
let's call $(a,b)$ by $d$, then:
$d|a, d|b$ so $xd|ax, xd|bx$ then $xd|(ax,bx)$
How can I continue from here? I didn't prove equality...
Continue on by writing out $xd\mid (xa,xb)$ as an equation and see what this says about the integers $\frac{(xa,xb)}{x}$, $a$, and $b$.