S is a set of N distinct positive integers such that no member of S has a prime factor greater than 35. Let P be the set of products of members of S taken 2 at a time. (For example, if x, y and z are members of S, then xy, xz and yz will be members of P.)

What is the smallest value of N for which it is certain that P contains a perfect square?

(In reply to Am I seeing all the possibilities? (solution, if I am) by Charlie)

I agree with your number Charlie. I was thinking that each number has to have a unique set of prime factors that are raised to an odd powered. With 11 primes, this gives 2^11=2048 choices. Notice that one of these choices is not having any primes to an odd power, which allows for the inclusion of exactly one perfect square in S.

Posted by owl on 2004-11-14 11:49:51