The King of Pointland

Properties of 2007

The number 2007 is divisible by three.

One way to tell whether a number is divisible by three is to add its digits together. If (and only if) that sum is divisible by three, then so is the original number. For example, for 2007, we note that 2 + 7 = 9, which is divisible by three. Therefore, so is 2007. In fact, 2007 = 3 x 3 x 223.

I don't remember when I learned this trick, but I was never taught why it works. From time to time I thought that I should figure it out, but I never got around to it. Until the other day. After a little thought, it became clear. The following example should give you the idea.

Consider the number 31512. 3+1+5+1+2=12, which is divisible by three, so 31512 is, too. This is why:

31512 = 3x10,000 + 1x1,000 + 5x100 + 1x10 + 2x1

= 3x(9,999 + 1) + 1x(999 + 1) + 5x(99 + 1) + 1x(9+1) + 2x1

= (3x9,999 + 1x999 + 5x99 + 1x9) + (3+1+5+1+2)

= 3x(3x3,333 + 1x333 + 5x33 + 1x3) + (3+1+5+1+2)

We always end up with a number of the form 3 x A + S, where S is the sum of the digits. If S is divisible by three, then S = 3 x C for some whole number C. So, we can write the number we're testing as 3 x A + 3 x C = 3 x (A+C), which means that the number is divisible by three.

Conversely, if we know the number is divisible by three, then 3 x A + S = 3 x K for some whole number K. So S = 3 x (K-A), which means that the sum of the digits, S, is divisible by 3.

Please don't tell me that you figured this out when you were ten, while waiting for the school bus. I already know.

Posted by cradle at January 3, 2007 05:44 PM