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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.

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