2015 AMC 12B Problem 3

Attempt Problem 3 of the 2015 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.

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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?

$8$

$11$

$14$

$15$

$18$

Answer: A

Concepts:Diophantine Equation divisibility

Difficulty rating: 1050

Solution:

Write $2x + 3y = 100.$ If $28$ were written twice, then $3y = 100 - 56 = 44,$ which is not a multiple of $3.$

So $28$ is written three times: $2x = 100 - 84 = 16,$ giving $x = 8.$

Thus, the correct answer is A.

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