Determining If a Large Number is Divisible by 11

Date: 10/22/2003 at 11:55:28
From: Concerned math student
Subject: 57986*11=637846??????

I just learned a trick to decide whether a large number is divisible
by 11 or not.  Here's an example to decide if 637846 is divisible by 11:

   637846

Cross out the last two digits (46) and add them to your remaining total:

   6378 + 46 = 6424

Cross out the last two digits (24) and add them to your remaining 
total.

   64 + 24 = 88

88 is divisible by 11, so the number 637846 is also divisible by 11.

Why does this method work?

Date: 10/22/2003 at 12:35:27
From: Doctor Peterson
Subject: Re: 57986*11=637846??????

Hi, Concerned.

This is related to the more familiar divisibility check listed in our 
FAQ:

  Divisibility Rules
    http://mathforum.org/dr.math/faq/faq.divisibility.html 

You can prove it by considering the rightmost two digits of a number 
(your 46) as a number y (less than 100), and the other digits (6378 
in your example) as another number x. Then the number you are starting
with is equal to:

  100x + y

When you take just the left part, and add to it the right part, you 
have:

  x + y

Now think about the difference between these two numbers:

  (100x + y) - (x + y) = (100x - x) + (y - y) = 99x

Since this difference is always a multiple of 11, then if one of the 
numbers is divisible by 11, so is the other. As a result, your
original number 100x + y is divisible by 11 if and only if the new, 
smaller number, x + y, is divisible by 11. Repeat the process until 
you get a number small enough to tell by sight whether it is.

If you have any further questions, feel free to write back.

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