## Problem 6

Find all positive integers $n$ such that $3^{n}+5^{n}$ is a multiple of $3^{n-1}+5^{n-1}$.
## Solution:

For such an $n$, since

$$
3\left(3^{n-1}+5^{n-1}\right)<3^{n}+5^{n}<5\left(3^{n-1}+5^{n-1}\right)
$$

it follows that

$$
3^{n}+5^{n}=4\left(3^{n-1}+5^{n-1}\right)
$$

which reduces to

$$
5^{n-1}=3^{n-1}
$$

implying $n=1$.

Since $n=1$ is a solution (because 8 is a multiple of 2), it is concluded that $n=1$ is the only solution.