Every once in a while you’ll need to test a number for divisibility by 11. Multiples of 11 can be quickly identified if the alternating sum of their digits is a multiple of 11. This works because the consecutive powers of 10 are either one more or one less than a multiple of 11. For example, 10 is one less than a multiple of 11, and 100 is one more than a multiple of 11, and 1000 is one less than a multiple of 11.

If you pull out these multiples of 11 from a number, then you are left with an alternating sum. Since the sum of 2 multiples of 11 is a multiple of 11, then the alternating sum is also a multiple of 11.

This is clearer in my notes, where a, b, c, and d are the digits of a 3 and 4 digit number:

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## Published by mathproblemsolvingskills

I coach students preparing for MathCounts and the AMCs, and I teach using curricula published by Art of Problem Solving.
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