I've been toying around with the following idea in my head for a while, and have never managed to come up with a counter-example, but I'm not sure if the following is true:

Suppose you have a list of positive integers. If one of the numbers in the list is divisible by a prime p, then there is at least one more number in the list that is also divisible by p. Also, if a different number is divisible by a prime q, then there is another number in the list divisible by pq. Can you always partition this list up into three smaller lists of no intersection such that if one partitioned list has an element divisible by a prime p, then in one of the other sets there is also an element divisible by p.

Guest Mar 6, 2021