number theory - $A$ must contain two relatively prime elements

Let $A$ be a set of positive integers such that no positive integer greater than $1$ divides all the elements of $A$. Prove or disprove that $A$ must contain two relatively prime elements.

This seems to be true, but I wasn't sure how to prove it. Maybe we can prove it by a proof by contradiction?

Answer

$\{6,10,15\}$ is a counterexample, or more generally $\{p_1p_2,p_1p_3,p_2p_3\}$ for any primes $p_1,p_2,p_3$.