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# Simon's Favorite Factoring Trick

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Given positive integers x and y such that x does not equal y and 1/x + 1/y = 1/18 , what is the smallest possible value for x+y ?

Any help would be appreciated. Thanks!

Jun 25, 2020

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Simplify: \(xy-18x-18y=0\)

Now we use Simon's Favorite Factoring Trick to solve:

Take out x from the first two terms: \(x(y-18)-18y=0\)

Add 18*18 to both sides: \(x(y-18)-18y+18*18=18*18\).

Factor again: \(x(y-18)-18(y-18)=18*18\).

Take out y-18: \((y-18)(x-18)=18*18\)

Since they can't be equal, the closest we can get is 27*12.

So y-18 = 27 and x-18 = 12

Now find x and y, add them up, and you have your answer.

Jun 25, 2020
edited by thelizzybeth  Jun 25, 2020
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amazingxin777  Jun 25, 2020
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Thank you! :)

thelizzybeth  Jun 25, 2020
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Thank You so much!!!

Guest Jun 25, 2020
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Ur Welcome

amazingxin777  Jun 25, 2020
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Hint:

x+y=xy/18

Now solve this! :D

18x+18y=xy

18x+18y-xy=0

18x+y(18-x)=0

Now solve!

y=18x/(x-18)

Jun 25, 2020