Problem:
Last year Isabella took math tests and received different scores, each an integer between and , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was . What was her score on the sixth test?
Answer Choices:
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Solution:
Let be the sum of Isabella's scores. Then is a multiple of , and
so is one of , or . Because is a multiple of , it follows that . Thus the sum of Isabella's first scores was , which is a multiple of , and the sum of her first scores was also a multiple of . Therefore her sixth score must have been a multiple of . Because her seventh score was and her scores were all different, her sixth score was . One possible sequence of scores is .
The problems on this page are the property of the MAA's American Mathematics Competitions