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, $lwh \geq 216$.