For which natural n is $3^n+5^n$ divisible by $3^{n-1}+5^{n-1}$? For which natural n is $3^n+5^n$ divisible by $3^{n-1}+5^{n-1}$? I only got that $3^n+5^n=8k$ when $n$ is odd. How to solve this one?

$$3(3^{n-1}+5^{n-1})=3^n+3*5^{n-1}<3^n+5^n<5*3^{n-1}+5^n=5(3^{n-1}+5^{n-1})$$
So the ratio is between $3$ and $5$, so it equals $4$.
$$4*3^{n-1}+4*5^{n-1}=3^n+5^n=3*3^{n-1}+5*5^{n-1}\\3^{n-1}=5^{n-1}$$
