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Integers: Even and Odd - Rules of Addition and Subtraction

If $$a$$ and $$b$$ are prime numbers, is $$ab$$ even? >(1) The sum of $$a$$ and $$b$$ is prime. >(2) The difference of $$a$$ and $$b$$ is prime.
Correct. [[snippet]] According to Stat. (1), $$a+ b$$ is a prime number. It is very easy to find examples where one of the numbers is 2 which satisfy the statement: Plug in $$a = 2$$ and $$b = 3$$: $$a+b$$ is a prime number (satisfying Stat. (1)), and $$ab$$ is even, giving an answer of "Yes" to the question stem. However, is it always "Yes"? Can you find an example of $$a$$ and $$b$$ where $$ab$$ is NOT even? For that, both $$a$$ and $$b$$ need to be odd; however, $$\mbox{odd}+\mbox{odd} = \mbox{even}$$, and there are no even primes other than 2. This means that it is not possible to find examples where $$a$$ and $$b$$ are both odd that satisfy Stat. (1). In other words, Stat. (1) tells you that $$a$$ and $$b$$ cannot be odd together, so one of them must equal 2, which means that the answer to the question stem is a definite "Yes." That's a definite answer, so **Stat.(1) → Yes → S → AD**. According to Stat. (2), $$a - b$$ is a prime number. * If you plug in $$a = 5$$ and $$b = 2$$, then $$a-b = 3$$, which is prime, so you can use these numbers. In this case, $$ab$$ is even. * If you plug in $$a = 5$$ and $$b = 3$$, then $$a-b = 2$$ , which is also prime, so you can still use these numbers. In this case, $$ab$$ is not even. No definite answer, so **Stat.(2) → Maybe → IS → A**.
Incorrect. [[snippet]] According to Stat. (1), $$a+ b$$ is a prime number. It is very easy to find examples where one of the numbers is 2 which satisfy the statement: Plug in $$a = 2$$ and $$b = 3$$: $$a+b$$ is a prime number (satisfying Stat. (1)), and $$ab$$ is even, giving an answer of "Yes" to the question stem. However, is it always "Yes"? Can you find an example of $$a$$ and $$b$$ where $$ab$$ is NOT even? For that, both $$a$$ and $$b$$ need to be odd; however, $$\mbox{odd}+\mbox{odd} = \mbox{even}$$, and there are no even primes other than 2. This means that it is not possible to find examples where $$a$$ and $$b$$ are both odd that satisfy Stat. (1). In other words, Stat. (1) tells you that $$a$$ and $$b$$ cannot be odd together, so one of them must equal 2, which means that the answer to the question stem is a definite "Yes." That's a definite answer, so **Stat.(1) → Yes → S → AD**.
Incorrect. [[snippet]] According to Stat. (2), $$a - b$$ is a prime number. * If you plug in $$a = 5$$ and $$b = 2$$, then $$a-b = 3$$, which is prime, so you can use these numbers. In this case, $$ab$$ is even. * If you plug in $$a = 5$$ and $$b = 3$$, then $$a-b = 2$$ , which is also prime, so you can still use these numbers. In this case, $$ab$$ is not even. No definite answer, so **Stat.(2) → Maybe → IS**.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
EACH statement ALONE is sufficient to answer the question asked.
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.