Consider the following statement. If n is any positive integer that is not a perfect square, then n is irrational. Fill in the blanks in the following proof of the statement. Proof: Suppose not. That is, suppose ---Select--- positive integer n such that n ---Select--- a perfect square and n is ---Select--- . By definition of ---Select--- there exist integers a and b such that n ? a b and b ? . By dividing a and b by all their common divisors if necessary, we may assume without loss of generality that a and b have no common divisor. Squaring both sides of n ? a b gives n ? a2 b2 , and multiplying by b2 gives nb2 ? a2. By the ---Select--- theorem, a, b, and n can be written as products of primes in ways that are unique except for the order in which the prime factors are written down. Now a2 and b2 are products of the same prime factors as a and b respectively, and, by ---Select--- each prime factor in a is written twice in a2, and each prime factor in b is written twice in b2. Thus, each prime factor in each