The question asks us to evaluate two statements about prime numbers and determine which are correct.

Statement Analysis

Statement 1: Every prime is odd

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. The sequence of prime numbers starts with 2, 3, 5, 7, 11, and so on.

• The number 2 is a prime number because its only divisors are 1 and 2.
• The number 2 is an even number.
• Therefore, the statement "Every prime is odd" is incorrect, as there exists at least one prime number (2) which is even.

Statement 2: Product of any two prime numbers is odd

Let's consider the product of two prime numbers. We need to check if the result is always odd.

• If we take two odd prime numbers, their product is odd (e.g., 3 × 5 = 15, which is odd).
• However, the prime number 2 is even. If we multiply 2 by any other prime number (odd or even), the result will be even.
• For example:
  • 2 × 3 = 6 (even)
  • 2 × 5 = 10 (even)
  • 2 × 2 = 4 (even)
• Since we found cases where the product of two prime numbers is even, the statement "Product of any two prime numbers is odd" is incorrect.

Conclusion

Based on the analysis, both Statement 1 and Statement 2 are incorrect.

Therefore, the correct option is that neither statement 1 nor statement 2 is correct.