Question 54
Let $M$ be the set of positive integers which is not divisible by any prime number greater than $3$. For arbitrary chosen subsets $A_1, A_2, A_3, \dots$ of $M$, prove that there exist two distinct positive integers $i$ and $j$ such that:
For each $x$ in $A_i$, $A_j$ has a divisor of $x$.
For each $x$ in $A_i$, $A_j$ has a divisor of $x$.