Problem:
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are members left over. The director finds that if they are arranged in a rectangular formation with more rows than columns, the desired result can be obtained. Find the maximum number of members this band can have.
Solution:
Let the square formation have rows and columns, and let the rectangular formation have columns and rows. Then , so . Because is a positive integer, , and there must be a positive integer for which . Then . Therefore or . Thus or , and the maximum number of members this band can have is .
The problems on this page are the property of the MAA's American Mathematics Competitions