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2021 Fall AMC 10B Problem 7

Problem 7 of 25EasierNumber TheoryArithmeticProblem-Solving Techniques

Call a fraction ab,\frac{a}{b}, not necessarily in simplest form, special if aa and bb are positive integers whose sum is 15.15. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

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Solution

A special fraction with denominator bb equals 15−bb=15b−1,\frac{15-b}{b}=\frac{15}{b}-1, where 1≤b≤14.1\le b\le14. We need integer values of 15x+15y−2.\frac{15}{x}+\frac{15}{y}-2. Taking x≤y,x\le y, a check of the fourteen possible denominators gives the following pairs that produce integers: (1,1),(1,3),(1,5),(2,2),(2,6),(2,10),(3,3),(3,5),(4,12),(5,5),(6,6),(6,10),(10,10). \begin{gathered} (1,1),(1,3),(1,5),(2,2),(2,6),\\ (2,10),(3,3),(3,5),(4,12),\\ (5,5),(6,6),(6,10),(10,10). \end{gathered} Their distinct sums are 1,2,3,4,6,7,8,13,16,18,28.1,2,3,4,6,7,8,13,16,18,28. There are 1111 such integers. Thus, the answer is C .
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Tagged: fraction · divisibility · systematic listing

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