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Washington and Lee University

Washington and Lee University
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February Problems

Easier Problem: Shelby has an unopened bag of candy hearts to share with her 16 classmates. When she distributes them evenly among herself and her friends, she has 3 pieces of candy left over. She decides to leave herself out so that the candy can be distributed evenly. So Shelby collects all the candy and begins to give it out again to her 16 classmates. When she does, she has 10 pieces left over. Sissy realizes that she is allergic to the candy and returns her pile to Shelby. So Shelby collects all the candy hearts again and distributes them among 15 of her classmates. This time, there are no candy hearts left over. What is the smallest number of candy hearts in the unopened bag?

Harder Problem: Consider the integers 724, 725 and 726. These consecutive integers are each divisible by a perfect square. Show that there are arbitrarily long strings of consecutive integers each divisible by a perfect square.