Problem:
Last year Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95 . What was her score on the sixth test?
Answer Choices:
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Solution:
Let be the sum of Isabella's 7 scores. Then is a multiple of 7 , and
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so is one of , or 679 . Because is a multiple of 6 , it follows that . Thus the sum of Isabella's first 6 scores was , which is a multiple of 5 , and the sum of her first 5 scores was also a multiple of 5 . Therefore her sixth score must have been a multiple of 5 . Because her seventh score was 95 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