2017 AMC 10B Problems/Problem 25
Contents
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?
Solution 1
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 .
Solution 1.1
First, we find the largest sum of scores which is which equals . Then we find the smallest sum of scores which is which is 7(98), 7(97), 7(96), 7(95)7(94)686, 679, 672, 665,65895591, 584, 577, 570,563570665706th5705957th\boxed{\textbf{(E) } 100}$.
==Solution 2 (Cheap Solution)==
By inspection, the sequences$ (Error compiling LaTeX. ! Missing $ inserted.)91,93,92,96,98,100,9593,91,92,96,98,100,95\boxed{\textbf{(E) } 100}$. 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.
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
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All AMC 10 Problems and Solutions |
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