If x and y are integers, then is x even?

Question

Directions: The question is followed by two statements. Answer using the codes given in the options.

If x and y are integers, then is x even?

Statement I: x²y² is even.

Statement II: 1 + x² + y² is odd.

1. A. The question can be answered using one of the statements alone, but cannot be answered using the other statement alone.
2. B. The question can be answered using either statement alone.
3. C. The question can be answered using both the statements together, but cannot be answered using either statement alone. (Correct answer)
4. D. The question cannot be answered even using both the statements together.

Correct Answer

Option C — The question can be answered using both the statements together, but cannot be answered using either statement alone.

Detailed Solution & Explanation

The correct answer is Option C (both together suffice, neither alone).

Key Points

• Statement I (x²y² even) means xy is even, i.e. at least one of x, y is even — but not necessarily x. Insufficient.
• Statement II (1 + x² + y² odd) means x² + y² is even, so x and y have the same parity — but that could be both odd. Insufficient.
• Together: same parity and at least one even forces both even, so x is even. Answer: Yes.

Additional Information

• Neither statement alone pins down x's parity, but the two combined do — hence Option C.