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?

Let $A_n=5^n+3^n,$
If $d$ divides $A_m,A_{m-1}$
$d$ will divide  $A_m-3A_{m-1}=2\cdot5^{m-1}$
$d$ will divide  $5A_{m-1}-A_m=2\cdot3^{m-1}$
So, $d$ will divide  $(2\cdot3^{m-1},2\cdot5^{m-1})=2(3,5)^{m-1}=2$
So, the necessary condition for $A_{m-1}\mid A_m$ is $: A_{m-1}$ must divide $2$
