We have 3^n + 5^n= 3(3^{n-1} +5^{n-1}) + 2*5^{n-1}
So if 3^{n-1} +5^{n-1} divides 3^n +5^n then it divides 2*5^{n-1}
But 3^{n-1} +5^{n-1} does not divide 5^{n-1} and they are co-primes
So 3^{n-1} +5^{n-1} divides 2
So 3^{n-1} +5^{n-1} = 1 or 2 so we get n= 1
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