Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$

Question

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$. Interested if there is a nice quick way other than mine.

Answer 1

Let $a_n=3^n+5^n$ then $3a_n\lt a_{n+1}\lt5a_n$ hence $a_n\mid a_{n+1}$ if and only if $a_{n+1}=4a_n$. The identity $a_{n+1}=4a_n$, in turn, is algebraically equivalent to $5^n=3^n$, that is, $n=0$.

Thus the only solution is $2=3^0+5^0\mid3^1+5^1=8$.

Answer 2

If $k(3^{n-1} + 5^{n-1} )= 3^n + 5^n$ then $k\leq 5.$