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This essential trains for: AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8, Math Kangaroo 9-10.
Again, the most useful tool is to write the number in expanded form:
and now write the powers of 10 using their closest multiple of 11:
Apply the distributive property of multiplication:
Apply the co-mutative property of addition to re-arrange the terms and factor an 11 out of the last terms, to emphasize the part that is a multiple of 11:
The largest part of the number is automatically divisible by 11. If the left over part which is the sum of the digits from right to left with alternating signs is also divisible by 11 then the whole number is divisible by 11.