2017 AMC 10B Problems/Problem 25

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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?

$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 94\qquad\textbf{(C)}\ 96\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 100$

Solution 1

Let the sum of the scores of Isabella's first $6$ tests be $S$. Since the mean of her first $7$ scores is an integer, then $S + 95 \equiv 0 \text{ (mod 7)}$, or $S \equiv 3 \text{ (mod 7)}$. Also, $S \equiv 0 \text{ (mod 6)}$, so by CRT, $S \equiv 24 \text{ (mod 42)}$. We also know that $91 \cdot 6 \leq S \leq 100 \cdot 6$, so by inspection, $S = 570$. However, we also have that the mean of the first $5$ integers must be an integer, so the sum of the first $5$ test scores must be an multiple of $5$, which implies that the $6$th test score is $\boxed{\textbf{(E) } 100}$.

Solution 1.1

First, we find the largest sum of scores which is $100+99+98+97+96+95+94$ which equals $7(98)$. Then we find the smallest sum of scores which is $91+92+93+94+95+96+97$ which is $7(94). So the possible sums for the 7 test scores so that they provide an integer average are$7(98), 7(97), 7(96), 7(95)$and$7(94)$which are$686, 679, 672, 665,$and$658$respectively.  Now in order to get the sum of the first 6 tests, we negate$95$from each sum producing$591, 584, 577, 570,$and$563$. Notice only$570$is divisible by$6$so, therefore, the sum of the first$6$tests is$570$. We need to find her score on the$6th$test so what number minus$570$will give us a number divisible by$5$. Since$95$is the$7th$test score and all test scores are distinct that only leaves$\boxed{\textbf{(E) } 100}$.

==Solution 2 (Cheap Solution)==

By inspection, the sequences$ (Error compiling LaTeX. ! Missing $ inserted.)91,93,92,96,98,100,95$and$93,91,92,96,98,100,95$work, so the answer is$\boxed{\textbf{(E) } 100}$. 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 (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Last Problem
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All AMC 10 Problems and Solutions

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