Question 199
Find, with reasons, all functions $f : \mathbb{N} \to \mathbb{N}$ such that
(i) $f(n) = 1$ if and only if $n = 1$,
(ii) if $d = \text{gcd}(n, m)$, then $f(nm) = \frac{f(n)f(m)}{f(d)}$, and
(iii) $f^{2004}(n) = n$ for every $n \in \mathbb{N}$, where $f^{2004}(n) = \underbrace{f(f(\cdots f(n)))}_{2004}$,
or show that they do not exist.
(i) $f(n) = 1$ if and only if $n = 1$,
(ii) if $d = \text{gcd}(n, m)$, then $f(nm) = \frac{f(n)f(m)}{f(d)}$, and
(iii) $f^{2004}(n) = n$ for every $n \in \mathbb{N}$, where $f^{2004}(n) = \underbrace{f(f(\cdots f(n)))}_{2004}$,
or show that they do not exist.