Question

Let \( p_1 \) and \( p_2 \) denote two arbitrary prime numbers. Which one of the following statements is correct for all values of \( p_1 \) and \( p_2 \)?

• A. \( p_1 + p_2 \) is not a prime number.
• B. \( p_1 p_2 \) is not a prime number.
• C. \( p_1 + p_2 + 1 \) is a prime number.
• D. \( p_1 p_2 + 1 \) is a prime number.

Hint:

Remember, the product of two distinct prime numbers is always composite. Understanding prime number properties is crucial for number theory problems.

Answer 1

The Correct Option is B

A prime number is defined as a number that has exactly two distinct positive divisors: 1 and itself.

For any two arbitrary prime numbers \( p_1 \) and \( p_2 \), their product \( p_1 p_2 \) will always have more than two divisors (i.e., 1, \( p_1 \), \( p_2 \), and \( p_1 p_2 \)), which means it cannot be a prime number.

Thus, the correct answer is (B).