Let p1 and p2 denote two arbitrary prime numbers. Which one of the following statements is correct for all values of p₁ and p₂?

The correct option is: p₁ × p₂ is not a prime number.

Step-by-step explanation:

To determine the correct statement for any two arbitrary prime numbers, let us denote them as:

$p_1$

and

$p_2$

Let us analyze the properties of prime numbers and their products.

1. Understanding Prime Numbers
By definition, a prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself.

2. Analyzing the Product

$p_1\times p_2$

Since both variables are prime numbers, they must be greater than or equal to 2 (the smallest prime number):

$p_1\ge 2$

and

$p_2\ge 2$

Now, let us examine the divisors of the product:

$p_1\times p_2$

The divisors include 1, the prime factors, and the product itself. Depending on whether the two prime numbers are distinct or identical, we have two cases:

Case A: The prime numbers are distinct

$p_1\ne p_2$

If the primes are distinct, the product has at least four distinct positive divisors: 1, the first prime, the second prime, and the product. Since it has more than two divisors, it cannot be prime.

Case B: The prime numbers are equal

$p_1=p_2=p$

If the primes are equal, the product is:

$p^2$

This has exactly three distinct positive divisors: 1, p, and p². Since it has more than two divisors, it cannot be prime.

Thus, in all cases, the product of any two prime numbers is a composite number (i.e., not a prime number).

3. Disproving other options:

- For the sum of two primes:

$p_1+p_2$

If we choose 2 and 3, their sum is 5, which is prime. Thus, the statement "p₁ + p₂ is not a prime number" is not always true.

- For the sum of two primes plus 1:

$p_1+p_2+1$

If we choose 3 and 5, their sum plus 1 is 9, which is composite. Thus, the statement "p₁ + p₂ + 1 is a prime number" is not always true.

- For the product of two primes plus 1:

$p_1\times p_2+1$

If we choose 3 and 5, the expression is 16, which is composite. Thus, the statement "p₁ × p₂ + 1 is a prime number" is not always true.