Remainder when a number is divided by 11

Last week we discussed how we can easily convert kilogram weights into pounds by using the technique to easily multiply by 11. We also used the divisibility rule for 11 to check the result of our multiplication. The logical continuation to that lesson is to explain how to use that rule to quickly determine the remainder when a number is divided by 11.

To check the result of multiplication by 11, we use the alternating sum check which can be done in two ways:

1. Alternately add and subtract the digits of the number and
2. Take the difference of the sums of alternating digits.

Now we will state the divisibility rule more clearly: a number is divisible by 11 if the difference of the sum of the digits in the odd position and the sum of the digits in the even position is 0 or a multiple of 11.

Obviously, if the difference is not 0 or 11, the number is not divisible by 11 and it will leave a remainder.

But how do we identify the odd or even positions? Do we start from the left of from the right? It does not matter if we have a number with an odd number of digits. But what about numbers having an even number of digits?

Let us look at 1243 which is exactly divisible by 11. (1 + 4) – (2 + 3) = 0. We then know that the next bigger 1244 will have a remainder of 1 when divided by 11.

If we start from the left, we will have (1 + 4) – (2 + 4) = – 1 while if we start from the right we will have (4 + 2) – (4 + 1) = 1 which is the correct remainder.

We can now state the remainder rule for 11: the remainder when a number is divided by 11 is the difference of the sum of the digits in the odd position starting from the right and the sum of the digits in the even position.

Let’s take some examples to see the remainders when divided by 11:

345: (5 + 3) – 4 = 2; 2 is the remainder.

765: (5 + 7) – 6 = 6; 6 is the remainder.

819: (9 + 8) – 1 = 16; we subtract 11 from 16 to get 5 the remainder.

Note that if we deduct 16 from 819, we will get 803 which is exactly divisible by 11 and 803 + 11 = 814 is also exactly divisible by 11. Thus 819 will leave a remainder of 5 (819 – 814 = 5).  

6251: (1 + 2) – (5 + 6) = – 8; add 11 to – 8 to get 3, the remainder. Here if we subtract 3 from 6251, we will get 6248 which is divisible by 11 since (8 + 2) – (4 + 6) = 0 .

Try to determine the remainder when the following numbers is divided by 11:

1. 41
2.  72
3. 364
4. 598
5. 844
6. 965
7. 2467
8. 8425
9. 21,456
10. 42, 986

Answers to last week’s exercises:

Multiply the following numbers by 11 and check your answers:

1. 53 x 11 = 583 (3 + 5) – 8 = 0
2. 72 x 11 = 792 (2 + 7) – 9 = 0
3. 57 x 11 = 627 (7 + 6) – 2 = 11
4. 85 x 11 = 935 (5 + 9) – 3 = 11
5. 352 x 11 = 3872 (2 + 8) – (7 + 3) = 0
6. 462 x 11 = 5082 (2 + 0) – (8 + 5) = – 11
7. 5216 x 11 = 57376 (6 + 3 + 5) – (7 + 7) = 0
8. 7429 x 11 = 81719 (9 + 7 + 8) – ( 1 + 1) = 22
9. 81354 x 11 =894,894 (4 + 8 + 9) – (9 + 4 + 8) = 0
10. 87659 x 11 = 964,249 (9 + 2 + 6) – (4 + 4 + 9) = 0