2004 AIME I Problems/Problem 1
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 37?
Solution
A brute-force solution to this question is fairly quick, but we'll try something more clever none the less: our numbers have the form , for
. Now, note that
so
, and
so
. So the remainders are all congruent to
. However, these numbers are negative for our choices of
, so in fact the remainders must equal
. Adding these numbers up, we get
, our answer.