Question: Suppose you know $G:=\gcd$ (greatest common divisor) and $L:=\text{lcm}$ (least common multiple) of $n$ positive integers; how many solution sets exist?
In the case of $n = 2$, one finds that for the $k$ distinct primes dividing $L/G$, there are a total of $2^{k-1}$ unique solutions.
I am happy to write out a proof of the $n = 2$ case if desirable, but my question here concerns the more general version. The $n=3$ case already proved thorny in my explorations, so I would be happy to see smaller cases worked out even if responders are unsure about the full generalization.
Alternatively: If there is already an existing reference to this problem and its solution, then a pointer to such information would be most welcome, too!