For all odd positive integer $n,$ the number $n\left(n^2-1\right)$ is divisible by

For $n = 1,$ we have 

$n\left(n^2-1\right)=0$ which is divisible by 24.

For $n = 3,$ we have

$n\left(n^2-1\right)=3\times (9-1)=24$ which is divisible by 24.

For $n = 5,$ we have

$n\left(n^2-1\right)=3\times (25-1)=72,$ which is divisible by 24.

Hence, $n\left(n^2-1\right)$ is divisible by 24.