Passage

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. (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement (b) Select this option if the question can be answered using either statement alone (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone (d) Select this option if the question cannot be answered even using any of the statements

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

(a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement ✓ UPSC's answer
(b) Select this option if the question can be answered using either statement alone
(c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone
(d) Select this option if the question cannot be answered even using any of the statements

Why the answer is (a)

• Statement I: xy² = 116 = 2² × 29. With y > 1, the only square factor available is 4, so y = 2 and x = 29, which is prime. Sufficient.

• Statement II: xz = 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. Not sufficient.

• One statement alone (I) answers the question and the other (II) does not: (a).

Why the other options are wrong

(b) Select this option if the question can be answered using either statement alone: Statement II alone is not sufficient: x can be 3 (prime) or 9 (not prime).
(c) Select this option if the question can be answered using both the statements together…: Statement I alone already answers the question.
(d) Select this option if the question cannot be answered even using any of the statements: Statement I alone gives x = 29, a definite answer.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.