$(1+1/a) (1+1/b) (1+1/c) = 4 $ Are there any other positive integer than 1,2,3 which satisfy the equation?

Question

$(1+1/a) (1+1/b) (1+1/c) = 4 $ Are there any other positive integers than $1,2,3 $ which satisfy the equation?

Answer 1

The answer is no; proof follows.

Assume without loss of generality that $a\le b\le c$. If $a\ge 2$, then the expression is at most $(3/2)^3<4$, a contradiction. Hence $a=1$, and we rewrite as $$(1+1/b)(1+1/c)=2$$ Now, if $b=1$, then $1+1/c=1$, a contradiction. Hence $b\ge 2$. But if $b\ge 3$, then $(1+1/b)(1+1/c)\le (4/3)^2<2$, a contradiction. Hence $b=2$. I leave it to you to solve for $c$ from here (only $c=3$ works).