29. A student says that if n is an odd integer greater than 1 and v is not a prime number, then there are always two prime numbers whose product is n. Which of the following integers disproves this statement?
Answer: D
45 disproves the statement regarding the existence of two prime numbers whose product is n.
The integer 45 serves as a counterexample to the claim that for any odd integer n greater than 1, if v is not a prime number, there will always be two prime numbers whose product equals n. Specifically, 45 can be expressed as the product of its prime factors 3 and 5, but it cannot be expressed as the product of two distinct prime numbers in a way that satisfies the condition of the statement.
A) 15
The integer 15 can be expressed as the product of the prime numbers 3 and 5. Since it meets the condition of being an odd integer greater than 1 and can be factored into two primes, it does not disprove the statement.
B) 21
The integer 21 can be factored into the prime numbers 3 and 7. It is also an odd integer greater than 1, and since it can be expressed as the product of two primes, it does not serve as a counterexample to the original claim.
C) 35
The integer 35 is the product of the prime numbers 5 and 7. Being an odd integer greater than 1, it fulfills the criteria set forth in the statement and does not disprove it, as it can indeed be expressed as a product of two primes.
D) 45
The integer 45 is equal to 9 multiplied by 5, where 9 is not a prime number; it can also be factored as 3 × 3 × 5. While it includes prime numbers, it cannot be represented as the product of exactly two distinct prime numbers, thus disproving the original assertion.
Conclusion
The integer 45 effectively disproves the statement because it cannot be represented as the product of two distinct prime numbers, despite being an odd integer greater than 1. In contrast, the other integers—15, 21, and 35—all can be expressed as the product of two prime numbers, thereby supporting rather than contradicting the assertion.