2015 AMC 10B Problem 3

Below is the professionally curated solution for Problem 3 of the 2015 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 10B solutions, or check the answer key.

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Concepts:linear equation divisibility casework

Difficulty rating: 870

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100,$ and one of the numbers is $28.$ What is the other number?

$8$

$11$

$14$

$15$

$18$

Solution:

Let the number written twice be $x$ , and let the number written three times be $y$ . Then $2x+3y=100$ .

If $x=28$ , then $3y=100-56=44$ , impossible for an integer $y$ . Therefore $y=28$ , and $2x=100-84=16$ , so $x=8$ .

Thus, the correct answer is A.

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