ISL 1989 Problem 96

Let $f : \mathbb{N} \mapsto \mathbb{N}$ be such that

(i) $f$ is strictly increasing; (ii) $f(mn) = f(m)f(n) \forall m, n \in \mathbb{N};$ and (iii) if $m \neq n$ and $m^n = n^m,$ then $f(m) = n$ or $f(n) = m.$

Determine $f(30).$