2017 AMC 10B Problems/Problem 25
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?
Let the sum of the scores of Isabella's first tests be . Since the mean of her first scores is an integer, then , or . Also, , so by CRT, . We also know that , so by inspection, . However, we also have that the mean of the first integers must be an integer, so the sum of the first test scores must be an multiple of , which implies that the th test score is .
First, we find the largest sum of scores which is which equals . Then we find the smallest sum of scores which is which is . So the possible sums for the 7 test scores so that they provide an integer average are and which are and respectively. Now in order to get the sum of the first 6 tests, we negate from each sum producing and . Notice only is divisible by so, therefore, the sum of the first tests is . We need to find her score on the test so what number minus will give us a number divisible by . Since is the test score and all test scores are distinct that only leaves .
Solution 2 (Cheap Solution)
By inspection, the sequences and work, so the answer is . Note: A method of finding this "cheap" solution is to create a "mod chart", basically list out the residues of 91-100 modulo 1-7 and then finding the two sequences should be made substantially easier.
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