Given the GCD and LCM of n positive integers, how many solutions are there? 
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!
 A: If you are interested in counting tuples $(a_1,a_2,\dots,a_n)$ such that $\gcd(a_1,\dots,a_n) = G$ and $\operatorname{lcm}(a_1,\dots,a_n) = L$ then we can do it as follows.
If $L/G = \prod\limits_{i=1}^s p_i^{x_i}$ then each  $a_i$ must be of the form $G \prod\limits_{j=1}^s p_i^{y_{i,j}}$ with $0 \leq y_{i,j} \leq x_i$.
Hence for each prime $p_i$ we require that the function from $\{1,\dots, n\}$ to $\mathbb N$ that sends $j$ to $y_{i,j}$ be a function that hits $0$ and $x_i$.
The number of such functions is easy by inclusion-exclusion for $x_i \geq 1$, it is $(x_i+1)^n - 2(x_i)^n + (x_i-1)^n$.
It follows the total number of tuples is $\prod\limits_{i=1}^s ( (x_i+1)^n - 2x_i^n + (x_i-1)^n)$.
A: (Adding this community wiki answer to point out a relevant reference.) I was recently pointed to the following paper, in which this and related problems are proposed and solved:

Bagdasar, O. (2014.) "On some functions involving the lcm and gcd of integer tuples." Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and mechanics, 6(2):91-100. PDF (no paywall).

The result appears as Theorem 2.7 (cf. the comment of Yorch, too):

