Find when $\frac{3^n+6^n+9^n}{3^{n-1}+6^{n-1}+9^{n-1}}$ is an integer 
Find all natural numbers $n$ such that $$\dfrac{3^n+6^n+9^n}{3^{n-1}+6^{n-1}+9^{n-1}}$$ is an integer.

For $n = 1$ we have $\dfrac{18}{3} = 6$.
For $n = 2$ we have $\dfrac{126}{18} = 7$.
For $n = 3$ we have $\dfrac{972}{126} = \dfrac{54}{7}$
For $n = 4$ we have $\dfrac{7938}{972} = \dfrac{49}{6}$.
I don't seem to find any other integers for the fraction for $n > 2$. How can we find all of the $n$ such that it is an integer?

Hint: It looks like considering the top and bottom $\mod 4$ narrows it down quite a bit.
