Problem:
Let be the set of positive integers with the property that the last four digits of are , and when the last four digits are removed, the result is a divisor of . For example, is in because is a divisor of . Find the sum of all the digits of all the numbers in . For example, the number contributes to this total.
Solution:
For any in , there are positive integers and such that . Simplifying, it is apparent that must be a factor of . The number has 12 divisors: , and . Each factor contributes the sum of its digits and the sum of the digits of . We can manually sum across the factors and add to find the requested sum, which is
The problems on this page are the property of the MAA's American Mathematics Competitions