Case 1: \(n\) is even. Then \(n = 2k\) for some integer \(k\text{.}\) This give

\begin{align*} n^3 - n \amp = 8k^3 - 2k\\ \amp = 2(4k^2 - k)\text{,} \end{align*}

and since \(4k^2 - k\) is an integer, this says that \(n^3-n\) is even.