Question 67
Let $P$ be the set of the first 2009 prime numbers and let $X$ be the set of all natural numbers that have only prime factors from $P$. Determine all natural numbers $k$ for which there exists a function $f: X \rightarrow X$ that satisfies the following equation for all $m, n \in X$: $$f(m f(n))=f(m) n^{k}$$