Let $a,b,c$ be elements in a GCD domain (or just a GCD monoid) $R$. Suppose that $\text{gcd}(a,b)=1$ and $c \ne 0$. If $R$ is a UFD, it is possible to write $c=a'b'$ with $\text{gcd}(a',a)=\text{gcd}(b',b)=\text{gcd}(a',b')=1$. Is this still true in any GCD monoid, or is there a counterexample?
This question is equivalent to a previously asked question, but I suspect it's easier to answer / find in the literature when asked in this form.
The question still make sense in an arbitrary commutative monoid $R$, where $\text{gcd}(a,b)=1$ can be interpreted as saying $a$ and $b$ are relatively prime, meaning all common divisors are units. However this generalized question (even if $R$ is assumed cancellative) does not obviously imply the generalized version of the equivalent question, which says that for all $a,b\in R$, there exist relatively prime elements $a' \mid a$ and $b' \mid b$ such that and both $a$ and $b$ divide $a'b'$.
