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2015 AMC 10B Problem 3

Problem 3 of 25EasierAlgebraNumber Theory

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?

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Solution

Let the number written twice be xx, and let the number written three times be yy. Then 2x+3y=1002x+3y=100. If x=28x=28, then 3y=10056=443y=100-56=44, impossible for an integer yy. Therefore y=28y=28, and 2x=10084=162x=100-84=16, so x=8x=8. Thus, the correct answer is A.

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Concepts: linear equation · divisibility · casework

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.