+434 Find the unique triple $(x,y,z)$ of positive integers such that $x < y < z$ and
\frac{1}{x} - \frac{1}{xy} - \frac{1}{xyz} = \frac{1}{3}
+135 \(\frac{1}{x} - \frac{1}{xy} - \frac{1}{xyz} = \frac{1}{3}\)
Multiply by x on both sides
\(1 - \frac{1}{y} - \frac{1}{yz} = \frac{x}{3}\)
x < 3
if x = 2:
\(\frac{1}{y}+\frac{1}{yz} = \frac{1}{3}\)
\(1 + \frac{1}{z} = \frac{y}{3}\)
Meaning z = 3, which cannot work in x > y > z
if x = 1:
\(\frac{1}{y}+\frac{1}{yz} = \frac{2}{3}\)
\(1 + \frac{1}{z} = \frac{2y}{3}\)
Also meaning z = 3, which doesn't work
Therefore, there is no solution.
