Dividing by 11-2

Tuesday, January 12, 2010

Any number written in our decimal system is made 
up of powers of 10. For example, 

   65,321 = 6(10^4) + 5(10^3)+3(10^2) + 2(10)+1. 

Now if you are interested in knowing whether 65,321 is a multiple of 5 
(that is, is exactly divisible by 5), you can tell just by looking 
at the last digit. That's because 10^4, 10^3, 10^2, and 10 are all 
divisible by 5, so the last digit on the end gets to cast the deciding 
vote.

Deciding whether 65,321 is a multiple of 3 is a little harder, since 
no power of 10 is a multiple of 3.  But 

         10 =  9   + 1 and
        100 = 99   + 1 and 
       1000 = 999  + 1 etc.

So if you had 65,321,

     65,321 = 6(10^4) + 5(10^3)+3(10^2) + 2(10)+1

you could rewrite this as

     65,321 = 6(9999+1) + 5(999+1) + 3(99+1) + 2(9+1)+1
            = 6(9999) + 5(999) + 3(99) + 2(9) + (6+5+3+2+1)

The first part is divisible by 3 so 65,321 is a multiple of 3 if the 
sum of its digits 6+5+3+2+1 is divisible by 3.

Now we are ready for 11.

Eleven is a little more complicated, but look at this neat pattern:

99                is a multiple of 11
9999              is a multiple of 11
999999            is a multiple of 11
...
any even number of 9's makes a multiple of 11.

If you look at how we wrote 
   
     65,321 = 6(9999+1) + 5(999+1) + 3(99+1) + 2(9+1) + 1

you can see that every other term contains a number that is a multiple 
of 11.  Unfortunately, numbers with an odd number of 9's are not 
divisible by 11.

But look at this pattern for these numbers:

10 = 10^1       = 11 - 1
1000 = 10^3     = 1001 - 1
100000 = 10^5   = 100001 - 1
10000000 = 10^7 = 10000001 - 1

and each of these numbers: 11, 1001, 100001, 10000001, ... is a 
multiple of 11. (Check it out by long division . . . it's worthwhile 
looking at the pattern you get when you divide 1 00 00 00 00 00 1 by 
11!)

Back to the example of 65,321.

    65,321 = 6(9999+1) + 5(1001 - 1) + 3(99+1) + 2(11-1) + 1

           = 6(9999) + 6 + 5(1001) - 5 + 3(99) + 3 + 2(11) - 2 + 1

All the terms with parentheses are multiples of 11. So 65,321 will 
be divisible by 11 if the remaining numbers are a multiple of 11. 
That is,
              6 - 5 + 3 - 2 + 1

Just the numbers you spoke about in your letter.

In general, the numbers in the odd positions are part of the "multiple 
of 999...999 + 1" and so are added to the total. The numbers in the 
even positions are part of the multples of "1000...0001 - 1" and so 
get subtracted.

The final test is to look at this alternating sum and difference of 
digits. But the result does not have to be 0 in order to be a multiple 
of 11...ANY multiple of 11 will do. For example,  

         8030209 gives 8 + 3 + 2 + 9 = 22.  

Since 22 is a multiple of 11, so is 8,030,209.