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2004 AMC 12A Problem 3

Problem 3 of 25EasierNumber TheoryCounting & Probability

For how many ordered pairs of positive integers (x,y)(x, y) is x+2y=100?x + 2y = 100?

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Solution

Writing x=1002y,x = 100 - 2y, the value of xx is a positive integer precisely when yy is a positive integer with 1y49.1 \le y \le 49. This gives 4949 valid ordered pairs. Thus, the correct answer is B.

More practice

Concepts: Diophantine Equation · counting integers in a range

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