Why does the divisibility rule for 11 work?

Answer 1

The rule comes about because of the place value system we use to write numbers in which each digit is worth ten times as much as when it is in the column to its right, for example:

• in 121 the 2 is worth 2 x 10 = 20 (twenty); whereas
• In 211 the 2 is worth 2 x 100 = 200 (two hundred) - tens times more than when it was in the second (tens) column.

Starting with the units column, each place value column is one more, one less, one more, one less, one more, etc than a multiple of 11, that is:

Units: 1 is 1 more than 0 = 0 x 11

Tens: 10 is 1 less than 11 = 1 x 11

Hundreds: 100 is 1 more than 99 = 9 x 11

Thousands: 1000 is 1 less than 1001 = 91 x 11

Ten Thousands: 10000 is 1 more than 9999 = 909 x 11

etc

So by alternately decreasing and increasing each place value column by its digit starting with the units column, it will produce a number which will definitely be a multiple of 11; however, to keep the value of the number the same, each digit will have to be added or subtracted from the number (depending upon whether the place value is decreased or increased):

Example: 12345

12345 = (1 x 10000) + (2 x 1000) + (3 x 100) + (4 x 10) + (5 x 1)

= (1 x 9999 + 1) + (2 x 1001 - 2) + (3 x 99 + 3) + (4 x 11 - 4) + (5 x 0 + 5)

= (1 x 9999 + 2 x 1001 + 3 x 99 + 4 x 11 + 5 x 0) + 1 - 2 + 3 - 4 + 5

= 11 x (1 x 909 + 2 x 91 + 3 x 9 + 4 x 1 + 5 x 0) + (1 - 2 + 3 - 4 + 5)

Thus the whole number can only be a multiple of 11 if the subtraction and addition of alternate digits (at the end above) is also a multiple of 11 (or zero), and hence the rule:

• A number is evenly divisible by 11 if when alternatively subtracting and adding the digits from the right hand end is a multiple of 11 or 0. Otherwise, the remainder of this sum divided by 11 is the remainder of the original number divided by 11.

This rule can also be expressed as:

• If the difference between the sum of the digits in the odd places and the sum of the digits in the even places is divisible by 11 (or 0), then the original number is divisible by 11.

In the case of 12345, 1 - 2 + 3 - 4 + 5 = 5 - 4 + 3 - 2 + 1 = 3, so 12345 is not a multiple of 11 (and has a remainder of 3).

If the remainder is negative, 11 needs to be added to find the remainder.

Examples:

• 15342

2 - 4 + 3 - 5 + 1 = -3

So 15342 is not a multiple of 11.

As -3 is negative, the remainder is -3 + 11 = 11 - 3 = 8.

(15342 = 11 x 1394 + 8)

• 231748

8 - 4 + 7 - 1 + 3 - 2 = 11

So 231748 is a multiple of 11.

(231748 = 11 x 21068)