$(1+1/x)(1+1/y)(1+1/z) = 3$ Find all possible integer values of $x$, $y$, $z$ given all of them are positive integers. Find all possible integer values of $x$, $y$, $z$ given all of them are positive integers
and 
$$(1+1/x)(1+1/y)(1+1/z) = 3.$$
I know
$(x+1)(y+1)(z+1) = 3xyz$ which is no big deal. I can't move forward now.
morever it is given $x$ is less than or equal to $y$ and $y$ is less than or equal to $z$

Let $x$ be the smallest of the three.
Case 1: $x=1$.
Then 
$$ y=\frac{2\,z+2}{z-2} = 2 + \frac{6}{z-2}$$
So $z-2$ should divide $6$
So the possibilities are $z=3,4,5,8$ If we want $x \le y \le z$, we can eliminate two of them.
case 2: $x=2$
Then
$$ y = \frac{z+1}{z-1} = 1 + \frac{2}{z-1}$$
So $z-1$ should divide 2, so z=2, or 3.
case 3:$x > 2$.
In this case show that $y < x$ and we can stop.
