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 $x + y$?
+128474 Let 18 = z
Let x = z + a and y = z + b
So we have
1/ (z + a) + 1/ (z + b) = 1 / z
(2z + a + b) / [ (z + a) (z + b) ] = 1 / z cross-multiply
(2z + a + b) z = (z + a) ( z + b)
2z^2 + az + bz = z^2 + az + bz + ab
z^2 = ab
Which means that 18^2 = ab = 324
a b x = z + a y = z + b
1 324 19 342
2 162 20 180
3 108 21 126
4 81 22 99
6 54 24 72
9 36 27 54
12 27 30 45
Smallest value of x + y = 30 + 45 = 75
