Let be a natural number. Prove that if , then is prime.
Suppose is composite; say , where the are pairwise coprime and greater than 1. (Say each is the highest power of some prime dividing .) If , each of the is represented in the set . Thus we have . If on the other hand is a prime power, and , then and are in the set , so that . If and , then , and . If , then .
So must be prime.