  We classify the finite groups with the property that any two different
character codegrees are coprime. In general, we conjecture that if $k$ is a
positive integer such that for any prime $p$ the number of character codegrees
of a finite group $G$ that are divisible by $p$ is at most $k$, then the number
of prime divisors of $|G|$ is bounded in terms of $k$. We prove this conjecture
for solvable groups.
