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 and solutions on this page are the property of the MAA's American Mathematics Competitions