Sums Divisible by 11

Date: 10/10/2001 at 20:13:42
From: Chris Stava
Subject: Why is the sum of an even-digit number and that same number 
in reverse always divisible by 11?

Hi Dr. Math,

You've been helpful before, so I'm calling on you again. 

A student's parent asked me why the sum of a number with an even 
number of digits (4, 6, 8...) and that same number written in reverse 
is always divisible by 11. (Example: 3456 + 6543)

I know the tricks to test for divisiblity, but I don't know WHY. Can 
you help?

I appreciate your time and effort. Thanks so much.
Sincerely,
Chris

Date: 10/10/2001 at 23:15:40
From: Doctor Peterson
Subject: Re: Why is the sum of an even-digit number and that same 
number in reverse always divisible by 11?

Hi, Chris.

Suppose the number has digits "ABCD," so it and its reverse "DCBA" are

    1000A + 100B + 10C + D  and  1000D + 100C + 10B + A

When we add them, we get

    1001A + 110B + 110C + 1001D

Since 110 = 11*10 and 1001 = 11 * 91, this is

    11*91 A + 11*10 B + 10 C + 11*91 D

This is clearly divisible by 11.

How about larger numbers of even digits? Then we have

    [10^(2n-1) A + ... + 10^0 Z] + [ 10^(2n-1) Z + ... + 10^0 A]

which becomes

    [10^(2n-1)+10^0]A + [10^(2n-2)+10^1]B + ...

Each pair of opposite digits (A and Z, B and Y, and so on) has a 
multiplier of the form

    10^(2n-1-k) + 10^k = 10^k (10^2(n-k)-1) + 1)

Recall that x^(2n+1) + 1 (an odd power of x plus 1) can be factored 
as (x+1) times another polynomial with integer coefficients (since 
x = -1 is a zero); so 10+1 is a factor of (10^2(n-k)-1) + 1).

This last part can be said more simply if you have to. Students may 
simply believe you when you say that the multipliers of pairs of 
digits will always be numbers like 11, 1001, 100001, and so on times a 
power of ten, and the former are all multiples of 11.

See the Dr. Math FAQ and our archives on the reason for the 
divisibility rule for 11, which is closely related.

   Divisibility Rules - Dr. Math FAQ
   http://mathforum.org/k12/mathtips/division.tips.html   

   Divisibility by 11 - Dr. Math archives
   http://mathforum.org/dr.math/problems/suria2.17.98.html   

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