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2014 AMC 12B Problem 7

Problem 7 of 25EasierAlgebraNumber Theory

For how many positive integers nn is n30−n\dfrac{n}{30-n} also a positive integer?

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Solution

Write n30−n=3030−n−1. \dfrac{n}{30-n} = \dfrac{30}{30-n} - 1. For this to be a positive integer, 30−n30-n must be a positive divisor of 3030 with 3030−n≥2,\dfrac{30}{30-n} \ge 2, i.e. 30−n≤15.30-n \le 15. The divisors of 3030 that are at most 1515 are 1,2,3,5,6,10,15,1, 2, 3, 5, 6, 10, 15, giving 77 values of nn (namely 15,20,24,25,27,28,2915, 20, 24, 25, 27, 28, 29). Thus, the correct answer is D.

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Concepts: divisibility · factor · algebraic manipulation

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