Question

Is the positive integer \(x\) odd? I) \(x^2\) is even.
II) \(4x\) is even.

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

Hint:

If \(x^2\) is even, then \(x\) is even. But \(4x\) is always even for every integer \(x\).

Answer 1

The Correct Option is A

Concept:
A positive integer is either odd or even. If \(x^2\) is even, then \(x\) must also be even.

Step 1: Check Statement I. Statement I says \[ x^2\text{ is even} \] If \(x\) were odd, then \(x^2\) would also be odd. Therefore, \(x\) cannot be odd. So \(x\) is even. This answers the question clearly: \(x\) is not odd. Hence Statement I alone is sufficient.

Step 2: Check Statement II. Statement II says \[ 4x\text{ is even} \] But \(4x\) is always even for every integer \(x\), whether \(x\) is odd or even. For example, \[ x=3\Rightarrow 4x=12 \] and \[ x=4\Rightarrow 4x=16 \] Both are even. So Statement II alone is not sufficient.

Step 3: Final answer. \[ \boxed{\text{Statement I alone is sufficient}} \]