Question

For the positive integer \( n \), is \( n - 10 \) odd?
I. \( n - 16 \) is a positive integer
II. \( n + 12 \) is even

• A. If the statement I alone is sufficient to answer the question.
• B. If the statement II alone is sufficient to answer the question.
• C. If the statements I and II together are sufficient to answer the question but neither statement alone is sufficient.
• D. If the statements I and II together are not sufficient to answer the question and additional data is required.

Hint:

When checking if an expression is odd or even, remember that:
- An even number minus an even number is even.
- An odd number minus an odd number is even.
- An even number minus an odd number is odd.

Answer 1

The Correct Option is B

We are asked if \( n - 10 \) is odd. To answer this, we need to use the given conditions. From the first condition, \( n - 16 \) is a positive integer. This means that: \[ n - 16 > 0 \quad \Rightarrow \quad n > 16 \] So, \( n \) must be greater than 16. From the second condition, \( n + 12 \) is even. For \( n + 12 \) to be even, \( n \) itself must be even, because an even number added to 12 (an even number) results in an even number. Thus, \( n \) is even. Now, we know that \( n \) is an even integer greater than 16. We need to check if \( n - 10 \) is odd. Since \( n \) is even, we have: \[ n - 10 = \text{even number} - 10 = \text{even number} \] Thus, \( n - 10 \) is even, not odd. Therefore, the answer is "No, \( n - 10 \) is not odd." Thus, the correct answer is \( \boxed{2} \).