Set of Integers with GCD of 1 are not necessarily Pairwise Coprime

Theorem
Let $S$ be a set of integers such that $S$ has more than $2$ elements:


 * $S = \set {s_1, s_2, \ldots, s_n}$

Let:
 * $\map \gcd S = 1$

where $\gcd$ denotes the GCD of $S$.

Then it is not necessarily the case that there exist a pair of elements of $S$ which are themselves pairwise coprime:
 * $\exists i, j \in \set {1, 2, \ldots, n}: \gcd \set {s_i, s_j} = 1$

Proof
Proof by Counterexample

Let $S = \set {6, 10, 15}$.

We have:

Hence the result.