Divisibility by 11
When you learned your divisibility rules, you learned simple rules for telling if a number could be divided by 2, 3, 4, 5, 9, and 10. Perhaps you learned a rule for 8. But there is also a simple way to tell if a number can be divided by 11.
If the difference of the sums of alternate digits is a multiple of 11, then the original number is a multiple of 11. For example: 162932. The sums of alternating digits are 1 + 2 + 3 = 6, and 6 + 9 + 2 = 17. The difference is 17- 6 = 11. So 162932 divides by 11. Another example: 565169. The sums of alternate digits are 5 + 5 + 6 = 16, and 6 + 1 + 9 = 16. The difference is 0. And 0 divides by 11 (0 times). So 565169 divides by 11.
Why does this work?
Consider some multiples of 10:
1 = 0 + 1
10 = 11 - 1
100 = 99 + 1 (99 = 9 x 11)
1000 = 1001 - 1 (1001 = 91 x 11)
10,000 = 9999 + 1 (9999 = 909 x 11)
100,000 = 100001 - 1 (100001 = 9091 x 11)
What this shows is that the multiples of 10 alternate between being one more and one less than multiples of 11. Each multiple of 10, when divided by 11, leaves a remainder of 1 or -1. Okay, we don't really do negative remainders. But it's the same as a remainder of 10.
So when we add alternating digits and find the difference of the results, we are really using a shortcut for finding the remainder when dividing by 11. And if that remainder is a multiple of 11, the number will divide by 11.
If the difference of the sums of alternate digits is a multiple of 11, then the original number is a multiple of 11. For example: 162932. The sums of alternating digits are 1 + 2 + 3 = 6, and 6 + 9 + 2 = 17. The difference is 17- 6 = 11. So 162932 divides by 11. Another example: 565169. The sums of alternate digits are 5 + 5 + 6 = 16, and 6 + 1 + 9 = 16. The difference is 0. And 0 divides by 11 (0 times). So 565169 divides by 11.
Why does this work?
Consider some multiples of 10:
1 = 0 + 1
10 = 11 - 1
100 = 99 + 1 (99 = 9 x 11)
1000 = 1001 - 1 (1001 = 91 x 11)
10,000 = 9999 + 1 (9999 = 909 x 11)
100,000 = 100001 - 1 (100001 = 9091 x 11)
What this shows is that the multiples of 10 alternate between being one more and one less than multiples of 11. Each multiple of 10, when divided by 11, leaves a remainder of 1 or -1. Okay, we don't really do negative remainders. But it's the same as a remainder of 10.
So when we add alternating digits and find the difference of the results, we are really using a shortcut for finding the remainder when dividing by 11. And if that remainder is a multiple of 11, the number will divide by 11.
