RO-NMO 2006 Problem 74

Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$ , $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$ , where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$ . Prove that the set $A$ has exactly two elements.

Marius Gherghu, Slatina