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 that every odd integer greater than 1 can be expressed as a product of two prime numbers.

The number 45 is an odd integer greater than 1 and cannot be expressed as a product of two prime numbers, as its prime factorization is 3^2 × 5, which involves a repeated prime factor.

A (15) YOUR ANSWER

The number 15 is an odd integer greater than 1 and can be expressed as a product of two prime numbers: 3 and 5. Therefore, it does not disprove the student's statement.

B (21) YOUR ANSWER

The number 21 is also an odd integer greater than 1, and it can be expressed as a product of two prime numbers: 3 and 7. Thus, it does not serve to disprove the student's claim.

C (35) YOUR ANSWER

The number 35 is an odd integer greater than 1, which can be factored into the product of two prime numbers: 5 and 7. Consequently, it does not contradict the assertion made by the student.

D (45) CORRECT

The number 45, while an odd integer greater than 1, is factored as 3 × 3 × 5, indicating that it includes a repeated prime factor rather than two distinct primes. This characteristic makes it impossible to express 45 as a product of two unique prime numbers.

Conclusion

In conclusion, the integer 45 effectively disproves the student's assertion that every odd integer greater than 1 can be expressed as a product of two prime numbers. While other integers like 15, 21, and 35 can be decomposed into two distinct primes, 45's reliance on a repeated prime factor highlights the limitations of the original claim. Thus, it serves as a critical example demonstrating the necessity of distinct primes in such products.

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 that every odd integer greater than 1 can be expressed as a product of two prime numbers.

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