Problem:
Let be the 79 -digit number that is formed by writing the integers from 1 to 44 in order, one after the other. What is the remainder when is divided by 45 ?
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Solution:
The remainder when is divided by 5 is clearly 4 . A positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9 . The sum of the digits of is , so must be a multiple of 9 . Then must also be a multiple of 9 , and the last digit of is 5 , so it is also a multiple of 5 . Thus is a multiple of 45 , and leaves a remainder of when divided by 45 .
The problems on this page are the property of the MAA's American Mathematics Competitions