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?

Rationale

45 disproves the statement about two prime numbers whose product is n.

The integer 45 is the product of the prime factors 3 and 5, but disagrees with the student's assertion that any odd integer greater than 1 can be expressed as the product of two primes when v is not a prime number.

A (15) YOUR ANSWER

The integer 15 is the product of two prime numbers: 3 and 5. This supports the student's assertion, as 15 can indeed be expressed as a product of two primes, thus does not disprove the statement.

B (21) YOUR ANSWER

The integer 21 is also the product of two prime numbers: 3 and 7. Similar to 15, this does not contradict the student's claim, as 21 can be factored into two primes, supporting the statement rather than disproving it.

C (35) YOUR ANSWER

The integer 35 is the product of the prime numbers 5 and 7. Like 15 and 21, this integer supports the student's assertion, as it can be expressed as a product of two primes, thus failing to disprove the statement.

D (45) CORRECT

The integer 45, however, can be expressed as 3 × 15, where 15 is not a prime number. This demonstrates that there exist odd integers greater than 1 that cannot be factored into two primes when v is not prime, effectively disproving the student's statement.

Conclusion

The assertion that every odd integer greater than 1 can be expressed as the product of two primes is invalid, as demonstrated by the case of 45. While integers like 15, 21, and 35 conform to this idea, 45 shows that it is possible to have an odd integer that cannot be expressed as the product of two primes, thereby disproving the student's claim.

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?

45 disproves the statement about two prime numbers whose product is n.

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