If x, y and z are integers, each greater than 1, then is x a prime number?

Question

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

If x, y and z are integers, each greater than 1, then is x a prime number?

Statement I: xy² = 116

Statement II: xz = 261

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

Correct Answer

Option A — The question can be answered using one of the statements alone, but cannot be answered using the other statement alone.

Detailed Solution & Explanation

The correct answer is Option A (one statement alone suffices).

Key Points

• Statement I (xy² = 116): since 116 = 2² × 29 and y > 1, the only square factor is 4, giving y = 2 and x = 29, which is prime — a definite "Yes". Sufficient.
• Statement II (xz = 261): 261 = 3² × 29, so x could be 3, 9, 29 or 87 — a mix of prime (3, 29) and non-prime (9, 87). Insufficient.

Additional Information

• Statement I settles it and Statement II does not, so the answer is Option A.