Find all possible positive integer $n$ such that $3^{n-1} + 5^{n-1} \mid 3^n + 5^n $ 
Find all possible positive integers $n$ such that $3^{n-1} + 5^{n-1} \mid 3^n + 5^n $.

I don't know how to start with. Any hint or full solution will be helpful.   

If $n= 1$ then we have $3^{n-1} + 5^{n-1} = 1+ 1 = 2|8 = 3^{n}+5^{n}$.  So that's one solution.  
Assume $n > 1$. 
$3^n + 5^n = 3*3^{n-1} + 5*5^{n-1} = 3*3^{n-1} + 3*5^{n-1} + 2*5^{n-1} = 3(3^{n-1} + 5^{n-1}) + 2*5^{n-1}$.
So $3^{n-1} + 5^{n-1}|2*5^{n-1}$
If $p$ is prime and $p|3^{n-1} + 5^{n-1}$ then $p|2*5^{n-1}$ then $p = 5$ or $p= 2$
If $p = 5$ then $p\not |3^{n-1} + 5^{n-1}$.  So $p = 2$ and $3^{n-1} + 5^{n-1} = 2^k$ but $2^k| 2*5^{n-1}$ means $k = 1$ so $3^{n-1} + 5^{n-1} = 2$. Which can only happen if $n = 1$.
So $n =1$ is only solution. 
