I tried using factorization of $a^n-b^n$ for odd $n$ in an attempt to work through to a situation where the factors are such that they cannot have n as a factor. But I reached nowhere. Here's how I proceeded -
a-b=4 and can be ignored.
The latter term is essentially odd and not divisible by any odd number till $9$ (easy to prove without getting into calculations).
However, for some arbitrary odd number $x=p^k$ where $p \ge 11$ is prime, I cannot say whether the sum of two terms is divisible by $x$ or not when the two terms are individually not divisible by $x$.