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# Given positive integers x and y such that x not = to y and 1/x + 1/y = 1/12, what is the smallest possible positive value for x + y?

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Given positive integers x and y such that x not = to y and 1/x + 1/y = 1/12, what is the smallest possible positive value for x + y?

Jul 29, 2016

#1
+16

$$\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$$

This is equivalent to

$$xy - 12x -12y = 0$$

Add 144 to both sides to get

$$(x-12)(y-12) = 144$$

It's not hard to check all of the ways to split 144 into a product of two integers. The best we can do without allowing $$x=y$$ is

$$(x-12)(y-12) = 144 = 9 \times 16$$

So $$x = 28$$ and $$y = 21.$$

Jul 29, 2016

#1
+16

$$\frac{1}{x} + \frac{1}{y} = \frac{1}{12}$$

This is equivalent to

$$xy - 12x -12y = 0$$

Add 144 to both sides to get

$$(x-12)(y-12) = 144$$

It's not hard to check all of the ways to split 144 into a product of two integers. The best we can do without allowing $$x=y$$ is

$$(x-12)(y-12) = 144 = 9 \times 16$$

So $$x = 28$$ and $$y = 21.$$

Guest Jul 29, 2016