Let $R$ be a GCD domain. This means that it is an integral domain in which any 2 elements have a GCD, and hence also an LCM.
Question:
Is it true that if $z$ is a GCD or LCM of $x$ and $y$ in $R$, then $z^n$ is a GCD or LCM of $x^n$ and $y^n$ for any nonnegative integer $n$?
Remember that GCDs and LCMs are only defined up to associates.
