The editorial of problem ABC370G: Divisible by 3 from last Atcoder Beginner Contest contains an interesting subproblem : How to count the number of arrays of length $$$L$$$ where product of all elements is $$$P$$$? The editorial mentions that this is a good exercise for Blue/Yellow coders, so I created a blog talking about this idea.
https://cfstep.com/training/tutorials/math/count-arrays-with-fixed-product/
To help you verify correctness, I also created a practice problem : https://codeforces.net/group/7Dn3ObOpau/contest/549784
The problems are untested, if you see any issues with the model solution, do let me know in the comments.
Haven't read the question nor the solution, but based on title:
prime factorize the number, let it be $$${p_1}^{a_1} \cdot {p_2}^{a_2} \cdot {p_3}^{a_3} ... \cdot {p_m}^{a_m}$$$.
The answer of the question is the number of ways to distribute $$$a_1$$$ number of $$$p_1$$$ into $$$L$$$ integers, then $$$a_2$$$ $$$p_2$$$s etc. which is the product of the number of ways to solve this equation $$$x_1 + x_2 + x_3 + \cdots + x_L = a_i$$$ for each i from 1 to m, where all $$$x_i$$$s are non-negative integers which has the solution of $$$C(a_i + L-1, L-1)$$$ where $$$C(n, r)$$$ denotes the number of ways to choose r out of n objects, unordered
So the general answer is: