Question 164
We denote by $\mathbb{N}=\{1,2,3, \ldots\}$ the set of natural numbers excluding zero and $\mathbb{N}^{*}=\{0,1,2,3, \ldots\}$ the set of natural numbers including zero. Find all the functions $f: \mathbb{N} \rightarrow \mathbb{N}^{*}$ that are growing, i.e. $f(n) \geq f(m)$ if $n\gt m$, and such that $f(n m)=f(n)+f(m)$, for all $n, m \in \mathbb{N}$.