Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$
Problem
Source: OIFMAT II 2012 day 1 p2 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat
Tags: number theory, functional, functional equation in N