# Math Help - Basic proof

1. ## Basic proof

Show that $n(n^2 - 1)$ is divisible by 24 when n is an odd number.

What I've got so far is a lemma that the square of any odd number is of the form 8q + 1.

So I kinda have... $n(8q + 1 - 1)=8nq$

This is where I got stuck... please help lol

(good to see LaTeX up )

2. Originally Posted by DivideBy0
Show that $n(n^2 - 1)$ is divisible by 24 when n is an odd number.

What I've got so far is a lemma that the square of any odd number is of the form 8q + 1.

So I kinda have... $n(8q + 1 - 1)=8nq$

This is where I got stuck... please help lol

(good to see LaTeX up )
One way...

$n(n^2 - 1) = n(n + 1)(n - 1) = (n - 1)n(n + 1)$

So we have the product of three consectutive numbers, starting with an even number. So one of the three numbers will be divisible by 3, one of the two even numbers will be divisble by 4, and the remaining even number will be divisible by 2. Thus the product is divisible by 2 * 3 * 4 = 24.

-Dan