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!
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.