For which strictly positive integers $n$ do we have $5^{n-1}+3^{n-1} \mid 5^{n}+3^{n}$?
If $5^{n-1}+3^{n-1} \mid 5^{n}+3^{n}$, then $5^{n-1}+3^{n-1} \mid 5^{n}+3^{n}-5\left(5^{n-1}+3^{n-1}\right)=-2 \cdot 3^{n-1}$. Thus, $5^{n-1}+3^{n-1} \leqslant 2 \cdot 3^{n-1}$, hence $5^{n-1} \leqslant 3^{n-1}$ and we deduce that $n=1$. Conversely, for $n=1,2$ divides 8 and $n=1$ is our only solution.