SCHWEITZER 1993 Problem 2

Let A be a subset of natural numbers and let k , r be positive integers. Suppose that for any r different elements selected from A , their greatest common divisor has at most k different prime factors. Prove that A can be partitioned into B and C , where any element of B has at most k + 1 different prime divisors and $$\sum_{n\in C} \frac{1}{n} <\infty$$