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?and b and b? v. 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. a² Squaring both sides of √n? ✓ 62 -gives n ? ✓ and multiplying by b2 gives nb2 ? ✓a². b By the ---Select--- Now a² and b² are products of the same prime factors as a and b respectively, and, by ---Select--- Thus, each prime factor in each of a² and b² occurs ---Select--- Since n is not a perfect square, ---Select--- Because all prime factors in b² occur ---Select--- Since nb² ? ✓a², a² contains a prime factor that occurs ---Select--- This contradicts the fact that every prime factor of a² occurs ---Select--- Hence, the supposition is ---Select---✓, and the given statement is ---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. ✓each prime factor in a is written twice in a2, and each prime factor in b is written twice in b². times. ✓an odd number of times. ✓times, it follows that the product nb² contains a prime factor that occurs an odd number of times. ✓ times. ✓ times.