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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

Best Answer 

 #1
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+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
avatar
+16
Best Answer

\(\frac{1}{x} + \frac{1}{y} = \frac{1}{12}\)

 

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

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