Difference between revisions of "2017 AMC 10B Problems/Problem 25"
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==Solution 1== | ==Solution 1== | ||
− | Let the sum of the scores of Isabella's first 6 tests be <math>S</math>. Since the mean of her first 7 scores is an integer, then <math>S + 95 \equiv 0 \text{ (mod 7)}</math>, or <math>S \equiv 3 \text{ (mod 7)}</math>. Also, <math>S \equiv 0 \text{ (mod 6)}</math>, so by CRT, <math>S \equiv 24 \text{ (mod 42)}</math>. We also know that <math>91 \cdot 6 \leq S \leq 100 \cdot 6</math>, so by inspection, <math>S = 570</math>. 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 | + | Let the sum of the scores of Isabella's first <math>6</math> tests be <math>S</math>. Since the mean of her first <math>7</math> scores is an integer, then <math>S + 95 \equiv 0 \text{ (mod 7)}</math>, or <math>S \equiv 3 \text{ (mod 7)}</math>. Also, <math>S \equiv 0 \text{ (mod 6)}</math>, so by CRT, <math>S \equiv 24 \text{ (mod 42)}</math>. We also know that <math>91 \cdot 6 \leq S \leq 100 \cdot 6</math>, so by inspection, <math>S = 570</math>. However, we also have that the mean of the first <math>5</math> integers must be an integer, so the sum of the first <math>5</math> test scores must be an multiple of <math>5</math>, which implies that the <math>6</math>th test score is <math>\boxed{\textbf{(E) } 100}</math>. |
==Cheap Solution== | ==Cheap Solution== |
Revision as of 11:36, 16 February 2017
Contents
[hide]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 .
Cheap Solution
By inspection, the sequences and work, so the answer is .
See Also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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