Problem:
Let be the -digit number that is formed by writing the integers from to in order, one after the other. What is the remainder when is divided by ?
Answer Choices:
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Solution:
The remainder when is divided by is clearly . A positive integer is divisible by if and only if the sum of its digits is divisible by . The sum of the digits of is , so must be a multiple of . Then must also be a multiple of , and the last digit of is , so it is also a multiple of . Thus is a multiple of , and leaves a remainder of when divided by .
The problems on this page are the property of the MAA's American Mathematics Competitions