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Question: find the divisibility of a 7-digit number by 11
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Well, the last number divisible by 11 before 7 digits is 999999, so the answer is
⌊9999999−99999911⌋=818181.⌊9999999−99999911⌋=818181.
More generally, if aa is divisible by 11 and b>ab>a, then the number of integers in the half open interval (a,b](a,b] divisible by 1111 is given by ⌊b−a11⌋⌊b−a11⌋, as you can verify by trying a few small cases, and then an induction argument should follow pretty easily.
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