Problem

Source: India TST 2023 Day 1 P2

Tags: algebra, functional equation, function



Let $g:\mathbb{N}\to \mathbb{N}$ be a bijective function and suppose that $f:\mathbb{N}\to \mathbb{N}$ is a function such that: For all naturals $x$, $$\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. $$ For all naturals $x,y$ such that $x|y$, we have $f(x)|g(y)$. Prove that $f(x)=x$. Proposed by Pulkit Sinha