 Solution

Factor the expression as $n^3-n=(n-1)n(n+1)$. One of three consecutive integers is divisible by $3$, so $3\mid n^3-n$. If $n$ is odd, its neighbours $n-1$ and $n+1$ are consecutive even integers. One is divisible by $4$ and the other by $2$, so their product is divisible by $8$. Since $3$ and $8$ are coprime integers, their product divides the expression:The argument also includes negative odd integers and the zero product at $n=\pm 1$.