Any set of integers that relatively prime in pairs are relatively prime.
Proof
Let \(\{a_1,a_2,...,a_k\}\) be a set of integers such that \((a_i,a_j)=1\) for any \(i,j \leq k\). Now, suppose for the sake of contradiction that the integers \(a_1,..,a_k\) are not relatively prime. Therefore, \((a_1,...,a_k)=d\) for some \(d \in \mathbb{Z}\). This means that \(d \mid a_1\) and \(d \mid a_2\). However, this implies that \((a_1,a_2) \geq d\) but this is impossible since \((a_1, a_2) = 1\). This is a contradiction. Therefore, we must have \((a_1,...,a_k) = 1\). \(\ \blacksquare\)