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.   

Let $d$ be a positive integer such that  $d$ divides both  $3^{n-1}+5^{n-1}$ and $3^{n}+5^{n}$.
then d divides $5\cdot (3^{n-1}+5^{n-1})-( 3^{n}+5^{n} )= 3^{n-1}\cdot 2$,   and 
$3\cdot(3^{n-1}+5^{n-1})-( 3^{n}+5^{n} )= -5^{n-1}\cdot 2$
Thus, as $\gcd(5,3)=1$, it follows that $d$ is either $1$ ou $2$. 
Since  $3^{n-1}+5^{n-1}$ divides  $3^{n}+5^{n}$, we must have 
$d=3^{n-1}+5^{n-1}=2$, which gives $n=1$.
