IZhO 2019 Problem 5

Let $n>1$ be a positive integer. A function $f: I \rightarrow \mathbb{Z}$ is given, where $I$ is the set of all integers coprime with $n$. ( $\mathbb{Z}$ is the set of integers). A positive integer $k$ is called a period of the function $f$ if $f(a)=f(b)$ for all $a, b \in I$ such that $a \equiv b(\bmod k)$. It is known that $n$ is a period of $f$. Prove that the minimal period of the function $f$ divides all its periods.