Directions for the next 5 (five) items : Each item in this section contains a question followed by two statements. Answer each item using the following instructions and mark your response on the Answer Sheet accordingly. Question : 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

The correct answer is Option A. To determine whether x is a prime number, we evaluate each statement using prime factorization.

Why Option A is correct: Statement I: xy² = 116. The prime factorization of 116 is 29 × 2². The problem states that x and y are integers strictly greater than 1. This means y² must be a perfect square factor of 116 that is greater than 1. The only perfect square satisfying this condition is 4 (where y = 2). Consequently, x must be 116 ÷ 4 = 29. Because 29 is unambiguously a prime number, Statement I alone is sufficient to definitively answer the question "Yes."

Statement II: xz = 261. The prime factorization of 261 is 3² × 29. The integer factors of 261 greater than 1 are 3, 9, 29, and 87. Thus, x could be 3 or 29 (which are prime numbers), or it could be 9 or 87 (which are composite numbers). Because Statement II yields ambiguous results regarding whether x is prime, it is not sufficient on its own.

Why the other options are incorrect:

• Option B: This is incorrect because Statement II cannot answer the question alone due to the multiple composite and prime possibilities for x.
• Option C: This is incorrect because Statement I is fully sufficient on its own; combining the statements is mathematically unnecessary.
• Option D: This is incorrect because Statement I successfully and independently provides a definitive answer to the question.

Takeaway: In Data Sufficiency questions involving integer multiplication, always find the prime factorization of the product. This reveals the exact mathematical constraints on the variables, especially when perfect squares are involved.