Let 3^{n-1}+5^{n-1}|3^n+5^n then there exists some positive integer k such that 3^n+5^n=k(3^{n-1}+5^{n-1})

if k >=5

k(3^{n-1}+5^{n-1})>=5(3^{n-1}+5^{n-1})=5.3^{n-1}+5.5^{n-1}>3^n+5^n

then k<=4

3^n+5^n=3.3^{n-1}+5.5^{n-1}>3(3^{n-1}+5^{n-1})

then k>=4 that is k=4

3^n+5^n=4(3^{n-1}+5^{n-1}) which gives

5^{n-1}=3^{n-1} which becomes impossible if n>1

then n=1 as we see 2|8.