# Thread: Divisibility by 9

1. ## Divisibility by 9

Prove that, for any positive integer $n$

$n(n-1)(n-2) - 6 \begin{bmatrix} \frac{n}{3} \end{bmatrix}$

is divisible by 9.

$\begin{bmatrix} x \end{bmatrix}$ denotes the integer part of $x$.

I wasn't sure how to go about this question, any help?

2. Originally Posted by CuriosityCabinet
Prove that, for any positive integer $n$

$n(n-1)(n-2) - 6 \begin{bmatrix} \frac{n}{3} \end{bmatrix}$

is divisible by 9.

$\begin{bmatrix} x \end{bmatrix}$ denotes the integer part of $x$.

I wasn't sure how to go about this question, any help?
There are various methods you can try. How about using induction? Another way is to solve it on a case by case analysis. Prove it for the three cases n = 3k, n = 3k +1 and n = 3k + 2. The latter method gets rid of the greatest integer function and makes the solution pretty clear