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:

$\exists i, j \in \set {1, 2, \ldots, n}: \gcd \set {s_i, s_j} = 1$

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

We have:

$\ds \gcd \set {6, 10}$

$=$

$\ds 2$

$\ds \gcd \set {6, 15}$

$=$

$\ds 3$

$\ds \gcd \set {10, 15}$

$=$

$\ds 5$

Hence the result.

$\blacksquare$

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.3$ The Euclidean Algorithm