Find the smallest positive integer $k$ for which there exist a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ with the following two properties:
(i) For all positive integers $m, n$ of the same colour, $f(m+n)=f(m)+f(n)$.
(ii) There are positive integers $m, n$ such that $f(m+n) \neq f(m)+f(n)$.

In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both (i) and (ii) the positive integers $m, n$ are not necessarily different.

Merlijn Staps, the Netherlands