Finding integral values of $3$ different variables.

Question

I'm stuck between a proof.
This is where I've reached

$(a+1)(b+1)(c+1)=4abc$
where $a,b,c$ are 3 distinct positive integers.

This will complete my proof

If I prove that
$(a+b+c)=abc$ from above

I know that it's true and answer is $(a,b,c)= (1,2,3)$ but I can't find those missing steps in between.

Thanks for the help.

Answer 1

Write $$ a={bc+b+c+1\over 3bc-b-c-1}$$ Since $a$ and $bc+b+c+1$ are positive so is $3bc-b-c-1$. Since $a\geq 1$ we have $$3bc-b-c-1\leq bc+b+c+1$$ This is equivalent to $$(b-1)(c-1)\leq 2$$ Can you finish?