OIM 2013 Problem 1

A set $S$ of positive integers is said to be channeler if for any three distinct numbers $a,b,c \in S$ , we have $a\mid bc$ , $b\mid ca$ , $c\mid ab$ .

a) Prove that for any finite set of positive integers $\{ c_1, c_2, \ldots, c_n \}$ there exist infinitely many positive integers $k$ , such that the set $\{ kc_1, kc_2, \ldots, kc_n \}$ is a channeler set.

b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.