Problem:
Mrs.Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , and . What was the last scores Mrs.Walter entered?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Note that the integer average condition means that the sum of the scores of the first students is a multiple of . The scores of the first two students must be both even or both odd, and the sum of the scores of the first three students must be divisible by . The remainders when , and are divided by are , and , respectively. Thus the only sum of three scores divisible by is , so the first two scores entered are and (in some order), and the third score is . Since is larger than a multiple of , the fourth score must be larger than a multiple of , and the only possible is , leaving as the score of the fifth student.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions