Because 11 is one more than 10, the divisibility test for 11 is related to the test for 9. Remember that each power of 10 is one more than a multiple of 9. Some powers of 10 are also one more than a multiple of 11. For example, 1 is (0 • 11) + 1, and 100 is (9 • 11) + 1.

Moreover, although 10 and 1,000 are not one more than a multiple of 11, they are one less than a multiple of 11. That is, 1,000 = (91 • 11) - 1, and 10 = (1 • 11) - 1. So what powers of 10 are one more than a multiple of 11? And what powers of 10 are one less than a multiple of 11?

The base ten blocks below represent the number 1,111:

To determine if 1,111 is divisible by 11, we express 1,111 as a sum:

1,000 + 100 + 10 + 1 = (1,001 - 1) + (99 + 1) + (11 - 1) + 1

This can be rewritten as:

(1,001 + 99 + 11) + (-1 + 1 - 1 + 1), or 11 • (91 + 9 + 1) + 0.

Thus, 1,111 is divisible by 11.

Knowing this leads to the divisibility rule for 11. Here's the rule: Find the sum of the digits indicating odd powers of 10 (e.g., 10¹, 10³, 10⁵, etc.) and the sum of the digits indicating even powers of 10 (e.g., 10⁰, 10², 10⁴, etc.). If the difference between these two sums is divisible by 11, then the number is divisible by 11. In our example, we have (-1 + 1 - 1 + 1) which yields 0, and 0 is divisible by 11.

There are divisibility tests for 7 as well, but the calculations involved take longer than dividing by 7! Note 3

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