Let $D$ be an integral domain and $a,b,x \in D$.
If $d$ is a greatest common divisor ( gcd ) of $a$ and $b$, is it true that $dx$ is a gcd of $ax$ and $bx$?
Note that $D$ is an integer domain, not a GCD domain. So this question is not the same as any of the following questions：1 2 3 etc. More specifically, the existence of gcd of $ax$ and $bx$ is not guaranteed.
Any insights are much appreciated.
BTW: To avoid ambiguity, I haven't used notations such as $(a,b)$ or $(ax,bx)$, which makes me verbose. You may use it at will.
This is not true. Take $D=\Bbb Z[\sqrt{-5}]$, $a=2,b=1+1\sqrt{-5},x=3$. Using the norm we see that $d=1$ is a gcd of $a,b$. However, $3d=3$ is not a gcd of $ax=6$ and $bx=3+3\sqrt{-5}$. This is because $1+\sqrt{-5}\mid ax,bx$ but $1+\sqrt{-5}\nmid 3$. (consequently $ax$ and $bx$ don't have a gcd.)