IMO 2003 shortlist, tough number theory problem

[Fm Jakaria]

Let p be a prime number, and A be a set of positive integers. The set A has the following property:
1) Suppose a set S of prime numbers is formed, so that for every q belongs to S; there exists at least one number belongs to A, say it x, such that q divides x. Again, no prime number r, does not belonging to S exists so that there exists a number y belongs to A, such that r divides y. Then S contains exactly p-1 elements.
2) Consider a subset D of A. D has at least one element, D is not necessarily a proper subset of A. Denote the product of all elements of D by P(D). Then no integer z exists so that z^p = P(D).

Denote the number of elements of A by K. For any fixed p, determine the maximum value of K.