When is a number divisible by 11?
Add the digits that are in the odd-numbered places in the number and do the same with the digits that are in the even-numbered places. If the difference of the results is 0, or divisible by 11 then the number is divisible by 11.
An example: is the number 12344321 divisible by 11? If you have read the story about mirror numbers then you already know that this number is divisible by 11.
We add the red digits together and we add the blue digits together:
1 2 3 4 4 3 2 1 gives 1 + 3 + 4 + 2 = 10 and 2 + 4 + 3 + 1 = 10. The difference is 10 − 10 = 0, so 12344321 is divisible by 11.
Another example: is the number 1358024679 divisible by 11?
1 3 5 8 0 2 4 6 7 9. Adding red and blue gives: 1 + 5 + 0 + 4 + 7 = 17 en 3 + 8 + 2 + 6 + 9 = 28. The difference is 28 − 17 = 11. The result is divisible by 11, so 1358024679 is divisible by 11.
Dividing by 11, with a floating point number as result
If you divide the numbers 1 through 10 by 11, then behind the comma there are always exactly two digits that repeat themselves infinitely. You find these digits by multiplying the numerator by 9. So
1/11 = 0,090909090909... (because 9 × 1 = 09)
2/11 = 0,181818181818... (because 9 × 2 = 18)
3/11 = 0,272727272727... (because 9 × 3 = 27)
4/11 = 0,363636363636... (because 9 × 4 = 36)
5/11 = 0,454545454545... (because 9 × 5 = 45)
6/11 = 0,545454545454... (because 9 × 6 = 54)
7/11 = 0,636363636363... (because 9 × 7 = 63)
8/11 = 0,727272727272... (because 9 × 8 = 72)
9/11 = 0,818181818181... (because 9 × 9 = 81)
10/11= 0,909090909090... (because 9 × 10 = 90)
Notice that the numbers form mirrored pairs: 09 and 90, 18 and 81, 27 and 72, and so on.
Now if you need to divide any number by 11, apply a long division. This is not difficult, you only need simple multiples of 11. Once you know what the remainder is, apply the trick above to get the decimal places! As an example, we calculate 9876/11
8
11 9876
88
10
89
11 9876
88
107
99
8
897
11 9876
88
107
99
86
77
9
The remainder is 9, so 9876/11 = 897,81818181...