Elementary Methods in Number Theory Exercise 1.4.9

Prove that $n^5-n$ is divisible by 30 for every integer $n$.

 

Proof:

\begin{equation}
n^5-n=(n-1)n(n+1)(n^2+1)
\end{equation}
\begin{equation}
6|(n-1)n(n+1)
\end{equation}
So we just need to prove that
\begin{equation}
5|(n-1)n(n+1)(n^2+1)
\end{equation}
When
\begin{equation}
n\equiv 0,1,2,3,4\mod 5
\end{equation},it is easy to verify that
\begin{equation}
5|(n-1)n(n+1)(n^2+1)
\end{equation}.Done.