A rectangular prism has **integer** edge lengths. Find all dimensions such that its surface area equals its volume.

My Solution:

Let the edge lengths be represented by the variables $l, w, h$

Then $$lwh = 2\,(lw +lh + wh) \implies lwh = 2lwh\left(\frac{1}{h} + \frac{1}{w} + \frac{1}{l}\right)$$

Dividing both sides of the equation by $lwh$ yields $$1 = 2\left(\frac{1}{h} + \frac{1}{w} + \frac{1}{l}\right)$$

Or, $$\frac{1}{h} + \frac{1}{w} + \frac{1}{l} = \frac{1}{2}$$

Though perhaps a bit unnecessary, I used some algebraic deduction and number theory to find all the possible ordered triple pairs for the dimensions of the rectangular prism in the cases where **all** the dimensions are the same and **two** of the dimensions are the same. 

My answers were: $(6,6,6),(5,5,10),(8,8,4),(12,12,3)$

I have a hunch that no ordered pair exists where all three values are distinct, but is there a way to rigorously prove this?

Note: By the AM-GM Inequality, $lwh \geq 216$.