Fix a number $n$. Can we color the positive integers using precisely $n$ colors in such a way that for any $m$, the numbers $m, 2m, \ldots, nm$ all receive different colors? The question was posed by Péter Pach about 10 years ago. To this day it remains open in general, although some cases are known. I will present a survey of known results and some other problems it leads to. This is joint work with Pach and my former master's student Tommy Chartier.