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.