The proof that i came up is as follows.
Assume any non prime positive integer${}> 1$ will have factors of either $2,3,5$ or a combination of them. Let $S$ be a set of sets where each set conatians a combination of $2,3,5$ $S = \{\{\}, \{2\}, \{3\}, \{5\}, \{2,3\}, \{2,5\}, \{3,5\}, \{2,3,5\}\}$ Since $|S|$ is $8$ and there are $9$ positive integers, by pigeonhole-principle, some two of them should share all of their prime factors that are less than or equal to $5$.
Is my reasoning correct? I am specially concered about whether my assumption of "any non prime positive integer $> 1$ will have factors of either $2,3,5$ or a combination of them" is correct
I'm now working on this problem. I've come to the conclusion that it is a false statement. The question says "Any set of positive integers", so what happens if the set consists of only prime numbers? In that case, no number shares any prime factor (as 1 itself is not prime). Unless I am missing something, I would think that this is at least one counterexample which proves the statement false.