Dividing by 11

Tuesday, January 12, 2010

Let's look at 352, which is divisible by 11; the answer is 32. 3+2 is 5; another way to say this is that 35 -2 is 33. Now look at 3531, which is also divisible by 11. It is not a coincidence that 353-1 is 352 and 11 × 321 is 3531. Here is a generalization of this system. Let's look at the number 94186565. First we want to find whether it is divisible by 11, but on the way we are going to save the numbers that we use: in every step we will subtract the last digit from the other digits, then save the subtracted amount in order. Start with

9418656 - 5 = 9418651     SAVE 5
     Then 941865  - 1 = 941864      SAVE 1
     Then 94186   - 4 = 94182       SAVE 4
     Then 9418    - 2 = 9416        SAVE 2
     Then 941     - 6 = 935         SAVE 6
     Then 93      - 5 = 88          SAVE 5
     Then 8       - 8 = 0           SAVE 8

Now write the numbers we saved in reverse order, and we have 8562415, which multiplied by 11 is 94186565.

---

Here's an even easier method, contributed by Chis Foren: Take any number, such as 365167484. Add the first, third, fifth, seventh,.., digits.....3 + 5 + 6 + 4 + 4 = 22 Add the second, fourth, sixth, eighth,.., digits.....6 + 1 + 7 + 8 = 22 If the difference, including 0, is divisible by 11, then so is the number. 22 - 22 = 0 so 365167484 is evenly divisible by 11.