2015 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2015 AMC 12B, 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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Concepts:Diophantine Equationdivisibility

Difficulty rating: 1050

3.

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100,100, and one of the numbers is 28.28. What is the other number?

88

1111

1414

1515

1818

Solution:

Write 2x+3y=100.2x + 3y = 100. If 2828 were written twice, then 3y=10056=44,3y = 100 - 56 = 44, which is not a multiple of 3.3.

So 2828 is written three times: 2x=10084=16,2x = 100 - 84 = 16, giving x=8.x = 8.

Thus, the correct answer is A.

Problem 3 in Other Years