Problem:
The digits of a positive integer are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when is divided by ?
Solution:
Let the digits of , read from left to right, be , and , respectively, where is an integer between and , inclusive. Then , where . Thus the requested sum is
There are seven such four-digit integers, the smallest of which is , whose remainder when divided by is . The seven integers form an arithmetic sequence with common difference , whose remainder when divided by is , so the sum of the remainders is .
The problems on this page are the property of the MAA's American Mathematics Competitions