Given positive integers $x$ and $y$ such that $x\neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{18}$, what is the smallest possible value for $x + y$?

Guest Jul 4, 2021

#1

#3**0 **

\(\frac{1}{x} + \frac{1}{y} = \frac{1}{18}\\ \frac{x+y}{xy} = \frac{1}{18}\\ xy=18x+18y\\ xy-18x-18y=0\\ (x-18)(y-18)=324\)

The pair of numbers that multiplies to 324 that are closest to each other but are not equal to each other are 12 and 27, so the answer is (18+12)+(27+18)

btw a very similar question was already answered here:

textot
Jul 4, 2021