Question 24
For every $n \in \mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ with the following properties:
(i) $d(f(x))=x$ for all $x \in \mathbb{N}$;
(ii) $f(x y)$ divides $(x-1) y^{x y-1} f(x)$ for all $x, y \in \mathbb{N}$.
(i) $d(f(x))=x$ for all $x \in \mathbb{N}$;
(ii) $f(x y)$ divides $(x-1) y^{x y-1} f(x)$ for all $x, y \in \mathbb{N}$.