# Math Help - Simple Proofs Problem

1. ## Simple Proofs Problem

I need to show that (n^2) - 1 is a multiple of 24. n is an odd number but not divisible by 3.

I have a general understanding of the problem, but i am getting tripped up on the not divisible by 3 part.

Any help would be much appreciated!

2. Hello, dlee426!

Prove that $N \:=\:n^2 - 1$ is a multiple of 24,
where $n$ is an odd number but not divisible by 3.

We have: . $n$ is of the form $3a \pm 1$, where $a$ is even.
. . That is, $a = 2b$ for some integer $b.$

Then: . $n \:=\:3(2b) \pm 1 \:=\:6b\pm 1$

And: . $N \:=\:n^2-1 \:=\:(6b\pm1)^2 - 1 \:=\:36b^2 \pm12b \:=\:12b(b\pm1)$

We see that $N$ is divisible 12.

Note that $b(b\pm1)$ is the product of two consecutive integers.
. . That is, one of them is even, the other odd.
Hence, the product $b(b\pm1)$ is divisible by 2.

Therefore, $N$ is divisible by 24.