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2003 AMC 10A Problem 24

Problem 24 of 25HarderNumber Theory

Sally has five red cards numbered 11 through 55 and four blue cards numbered 33 through 6.6. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

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Solution

Among blue cards 3,3, 4,4, 5,5, 6,6, red 55 divides only 55 and red 44 divides only 4,4, so those pairs must sit at the ends. Red 22 divides only 44 and 6,6, and red 33 divides only 33 and 6.6. Chaining these forces the stack R4,R4, B4,B4, R2,R2, B6,B6, R3,R3, B3,B3, R1,R1, B5,B5, R5.R5. The middle three cards are B6,B6, R3,R3, B3,B3, summing to 6+3+3=12.6 + 3 + 3 = 12. Thus, the correct answer is E.

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Concepts: divisibility · logical deduction

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