(EGMO 2017) Find the smallest positive integer $k$ for which there exists a $k$-coloring of the positive integers and a function $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that

1. $\forall n, m \in \mathbb{N}^{*}$ of the same color, $f(n+m)=f(n)+f(m)$
2. $\exists n, m \in \mathbb{N}^{*}$ such that $f(n+m) \neq f(n)+f(m)$