2016/073) $\frac{1}{x} + \frac{1}{y} +\frac{1}{z}=1$ what is the value of $x,y$ and $z$ if $x>y>z$ and they are integers

$\frac{1}{x} + \frac{1}{y} +\frac{1}{z} > 3\frac{1}{z} = \frac{3}{z} = 1$

hece $ z < 3$ and as z cannot be 1 so $z = 2$
hence $\frac{1}{x} + \frac{1}{y} =\frac{1}{2}$
hence $2y+ 2x = xy$
hence $xy-2y-2x = 0$
or $xy-2x-2y + 4 = 4$
or $(x-2)(y-2) = 4$
as $x> y$ so we get $x = 2 = 4, y=2 =1$ giving $x = 6, y= 3$
hence $z=2,y=3,x = 6$