5. Let $n$ be a certain positive integer. When substituting $n$ into the algebraic expression $n^{5}-n$ to calculate its value, four students came up with the following four results, of which only one is correct. Then this correct result is ( ).
()) 777 ()) 777 ()) 1776 ()) 7779
5.A.

Since $I=n^{5}-n=(n-1) n(n+1)\left(n^{2}+1\right)$, it is clear that $I$ is a multiple of 2, thus eliminating options (B) and (D). When $n=5k, 5k+1, 5k-1$, $I$ is a multiple of 5; when $n=5k-2, 5k+2$, $n^{2}+1$ is a multiple of 5. Therefore, $I$ is a multiple of 5. Hence, regardless of what positive integer $n$ is, $I$ is always a multiple of 5, eliminating option (C).