Solve a system of diophantine equations

Question

I have a problem with elemental number theory. I started with the expression $$ (a - \frac{1}{b})(b - \frac{1}{c})(c - \frac{1}{a}) $$

and task to find all natural $a,b,c$ so that the result of the expression is an integer.

I managed to show that-

$$ a | bc - 1$$

$$b | ac - 1$$

$$c | ab - 1 $$

But I don't know how to solve this 3 equations. Is there even enough data?

Answer 1

Expand to get $abc|ab+ac+bc+1$.

Then $\frac{1}{a}+\frac{1}b+\frac{1}c+\frac{1}{abc}$ must be an integer. If $3 \le a \le b \le c$,

then $\frac{1}{a}+\frac{1}b+\frac{1}c+\frac{1}{abc} \le \frac{28}{27}$. Now easy casework. ;)