Problem

Source: 2018 Taiwan TST Round 1

Tags: function, algebra, functional equation



Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$holds for all positive integers $ m,n $