  For which values of $n$ can we color the positive integers with precisely $n$
colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all get
different colors? Pach posed the question around 2008-9. Particular cases
appeared in K\"oMaL in April 2010, and the general version appeared in May 2010
on MathOverflow, posted by P\'alv\"olgyi. The question remains open. We discuss
the known partial results and investigate a series of related matters
attempting to understand the structure of these $n$-satisfactory colorings.
  Specifically, we show that there is an $n$-satisfactory coloring whenever
there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$
compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are
in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases
where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or
there is a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct
modulo $q$ (in which case we call $q$ a strong representative of order $n$).
The colorings obtained by this process we call multiplicative. We also show
that nonmultiplicative colorings exist for some values of $n$.
  There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there
is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all
$n$-satisfactory colorings for $n\le 5$ and all multiplicative colorings for
$n\le 8$, and show that there are as many nonmultiplicative colorings of $K_n$
as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong
representative $q$ then the set of such $q$ has positive natural density in the
set of all primes.
  We show that the question of whether there is an $n$-satisfactory coloring is
equivalent to a problem about tilings, and use this to give a geometric
characterization of multiplicative colorings.
