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IMO ShortList 2003, number theory problem 8

Source: IMO ShortList 2003, number theory problem 8

October 4, 2004

modular arithmeticnumber theoryprime numbersPerfect PowersIMO Shortlist

Problem Statement

Let

p

be a prime number and let

A

be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in

A

consists of

p-1

elements;
(ii) for any nonempty subset of

A

, the product of its elements is not a perfect

p

-th power.
What is the largest possible number of elements in

A