Understand the problem

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$for all positive integers $a,b$.

Source of the problem

APMO 2019

Topic
Number Theory
Difficulty Level
Medium
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

Considering prime numbers might be a good idea.
Show that, for any prime p, f(p)|p^2. Hence or otherwise show that f(p)=p.
If gcd(p,b)=1 for a prime p, then show that p+b|p+f(b). This is equivalent to p+b|f(b)-b. Note that the RHS is independent of p.
As in p+b|f(b)-b the RHS is fixed and the LHS is arbitrarily large, the RHS has to be zero. Hence f(b)=b for every natural number b.

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