Suppose $f(n)=q^n$, where $q$ is an integer $>2$. (In the original formulation, $q$ should have been a prime, but I think it doesn't really matter)
Define the function $$ g(n) = \sum_{d|n} \mu\left(\frac nd \right)f(d) $$ that, by moebius inversion, is equivalent to $$ f(n) = \sum_{d|n} g(d). $$
Is it true that $n|g(n)$? I verified it for primes and powers of primes. Is it true in general? Probably the way to go is to apply the multiplicative property.