Three Consecutive Natural Numbers

Date: 09/17/2001 at 01:03:37
From: Jordan Harvey
Subject: Three Consecutive Natural Numbers

Why is it that the product of any three consecutive natural numbers 
is always divisible by six?

Date: 09/17/2001 at 10:01:43
From: Doctor Sarah
Subject: Re: Three Consecutive Natural Numbers

Hi Jordan - thanks for writing to Dr. Math.

Let's start with what we know about the divisibility rule for 6:

  If a number is divisible by both 2 and 3, it is also divisible by 6.

(See "Divisibility Rules," from the Dr. Math FAQ:
   http://mathforum.org/dr.math/faq/faq.divisibility.html    .)

Now let's look at the natural numbers:

   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...

 - Every other number (2 4 6 8...) is even, and is therefore 
   divisible by 2.

 - Every third number (3 6 9 12...) is a multiple of 3, and is 
   therefore divisible by 3.

So every combination of three consecutive natural numbers will include 
one even number and one multiple of three, and the product will 
therefore be divisible by 2 and 3, and so also by 6.

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/