Directions for the following 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
Answer & explanation
Answer: (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
Factorise both numbers: 116 = 2² × 29 and 261 = 3² × 29. The first leaves only y = 2 and x = 29, a prime; the second allows x = 3, 9, 29 or 87, so it cannot decide.
- 116 = 2² × 29. With y > 1, the only square factor is y² = 4, so y = 2 and x = 29.
- 29 is prime, so Statement I alone answers the question (yes).
- 261 = 3² × 29. With x, z > 1, x can be 3, 9, 29 or 87.
- 3 and 29 are prime but 9 and 87 are not, so Statement II alone cannot answer.
- One statement alone works and the other does not.
- ✓ Statement I Sufficient alone: 116 = 4 × 29 forces y = 2 and x = 29, which is prime.
- ✗ Statement II Not sufficient alone: 261 = 9 × 29 allows x = 3 or 29 (prime) and x = 9 or 87 (not prime).
Remember · Prime factorisation first; then ask which factor splits are allowed by the given conditions (here, every variable > 1).
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·