Let $x$ be a whole number. Show that $x(x-1)(x+1)$ divided by $6$ is a whole number.

Question

Proving things in mathematics is very difficult for me. I don't quite understand what to do here:

Let $x$ be a whole number Show that $x(x-1)(x+1)$ divided by 6 is a whole number

I really appreciate any help

Answer 1

What you need to do is show that $x(x+1)(x-1)$ is always a multiple of $6$. This is the same as showing that it is always a multiple of $2$ and it is always a multiple of $3$. Try what happens a few values of $x$ - can you see a pattern that tells you why $x(x+1)(x-1)$ is a multiple of $2$ in each case? What about $3$?

Answer 2

Note that $x-1,x,x+1$ are three consecutive numbers. Hence at least one of them is even (is a multiple of 2) and exactly one of them is multiple of 3. Hence their product is a multiple of 6.

Answer 3

This question tests your understanding of the integers modulo 6, also called $\mathbb{Z}_6$

You need to show that $$ x(x-1)(x+1)=0~ (mod ~6) $$ If $x=0$, $x=1$, or $x=5$, it’s true. If $x=2$ or $x=3$, we also have a multiple of 6, so it’s also true. Likewise if $x=4$, we have a multiple of 12, so it’s also true.