Show that $n(n^2-1)$ is divisible by 24, if $n$ is an odd integer greater than $2$. How can I show that $n(n^2-1)$ is divisible by 24, if $n$ is an odd integer greater than $2$?
I am thinking that since odd numbers have the form of $2n-1$ in which if it is to be more than $2$, it will be $2n-1+1 = 2n+1$. So would it be correct to use this and try solving through induction?

Let $n=2m+1$.
Among $2m(2m+1)(2m+2)$, one of the extreme factors is a multiple of $4$ and the other is even, and one of the three factors is a multiple of $3$.

Check with
$$2\cdot3\cdot4,\\4\cdot5\cdot6=4\cdot5\cdot2.3,\\6\cdot7\cdot8=2.3\cdot7\cdot2.4,\\8\cdot10\cdot12=2.4\cdot10\cdot2.2.3,\\\cdots$$
