Find tuples $(a,b)$

Question

(Second time I ask this because the previous was not very clear so I deleted it) Hi, I've trying to solve this problem (competitive programming) called Divide a days ago, it's states:

Given $p,q,n \le 10^{10}$, find all the tuples such that:

$\frac {ab} {a+b} = c $

($a+b|ab$, $a \in [1;p]$, $b\in [1;q]$, $c\in [1;n]$, all integers)

Note: this is not about to check whether $ab$ is divisible by $a+b$ but to find how many valid tuples (a,b) when put into that expression above, results results in a number $c\le n$

As the values $p,q,n$ are too big and cannot be calculated by brute-force, is there a way to calculate how much of such tuples exists in $O(1)$?

Original problem: https://dmoj.uclv.edu.cu/problem/2023orcop1112b

What I've reached by myself:

1. When $a=b$ then $\frac a 2 \le n$ and $a \le 2n$ , so that gives some tuples of the form $(2k, 2k)$

2. When $a=kb$ then $\frac {kb} {k+1}$ as $k$ and $k+1$ are coprime, must find a $k+1$ that divides $b$, (can be easily calculated)

Nothing more useful :/

Answer 1

A quick google search yielded some good explanations of this problem:

Necessary and sufficient conditions for the sum of two numbers to divide their product

https://codereview.stackexchange.com/questions/2327/find-the-no-of-pairs-whose-sum-divides-the-product-from-0-1000000000

In general, I think that with problems like this, it is often a good idea to split up terms into common factors and relatively prime integers (i.e., $a = a'g, b = b'g,$ for $a', b'$ relatively prime).