Problem

Source: CSMO 2019 Grade 10 Problem 3

Tags: function, number theory



Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a function such that $f(ab)$ divides $\max \{f(a),b\}$ for any positive integers $a,b$. Must there exist infinitely many positive integers $k$ such that $f(k)=1$?