Find the smallest number of positive integer divisors of $a+b$ knowing that $k=\frac{ab+c^2}{a+b}$

Question

Consider the positive integers $a, b, c$ such that the fraction

$k=\frac{ab+c^2}{a+b}$

Is a positive integer less than $a$ and $b$. Find the smallest number of positive integer divisors of $a+b$

Answer 1

As mentioned in comments, there's a solution with 3 divisors and it suffices to prove that $a+b$ cannot be prime. Suppose that $a+b=p$ is prime. We have the condition $c<a,b$ (equivalent to the given constraints). We also have $p|c^2-a^2$, which implies that $p|(c-a)(c+a)$. This is now a contradiction as $|c-a|$ and $(c+a)$ are both less than $p$.