Part B

Question

James says that if a prime number is doubled, then that new number is still a prime number.

State whether you agree or disagree, and explain why you think so (e.g. you may refer to a definition). If you agree, give an example that proves James's conjecture. If you disagree, give a counterexample that disproves James's conjecture.

(2 points)

Answer 1

I disagree with James's statement that if a prime number is doubled, then the new number is still a prime number.

A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. When a prime number is doubled, the resulting number is even and greater than 2, which means it can be divided by 2, making it a composite number rather than a prime.

For example, consider the prime number 3. Doubling it gives us \(3 \times 2 = 6\), which is not prime because it can be divided by 1, 2, 3, and 6. Therefore, 6 has divisors other than just 1 and itself, disproving James's conjecture.

Thus, doubling a prime number does not result in a prime number.