Problem 26 · 2011 Math Kangaroo
Stretch
Number Theory
factorizationfactor-pairs
How many ordered pairs of positive whole numbers \((x, y)\) solve the equation \(\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{3}\)?
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Answer: D — 3
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Hint 1 of 2
Clear the fractions and rearrange into a product of two factors equal to a fixed number.
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Hint 2 of 2
Then count the positive factor pairs of that number.
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Approach: turn it into (x-3)(y-3) = 9
- From 1/x + 1/y = 1/3, clearing denominators gives 3y + 3x = xy, i.e. (x−3)(y−3) = 9.
- Positive solutions need x−3 and y−3 to be a positive factor pair of 9: (1,9), (3,3), (9,1).
- These give (x,y) = (4,12), (6,6), (12,4).
- So there are 3 ordered pairs, choice (D).
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