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 that 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 we double a prime number, the resulting number will typically be even (except in the case of the prime number 2).

For example, let's consider the prime number 3. When we double it, we get:

\( 2 \times 3 = 6 \)

The number 6 is not a prime number because it has divisors 1, 2, 3, and 6. Thus, it is not true that doubling a prime number results in another prime number.

In fact, the only even prime number is 2, and when we double it:

\( 2 \times 2 = 4 \)

The number 4 is also not prime (it has divisors 1, 2, and 4).

Thus, the claim that doubling a prime number will yield another prime number is false, as demonstrated by the counterexamples above.