UPSC Prelims 2026 CSAT Paper II · Q5 of 70 Comprehension medium

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

  1. (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
  2. (b) Select this option if the question can be answered using either statement alone
  3. (c) Select this option if the question can be answered using both the statements together, but cannot be answered using either statement alone
  4. (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. Since x and y are integers > 1, y² must be a perfect square divisor of 116. The divisors of 116 are 1, 2, 4, 29, 58, 116. The only perfect square > 1 is 4. Thus y² = 4, so y = 2. Then x = 116/4 = 29. Since 29 is a prime number, the answer is 'Yes'. Statement I alone is sufficient.

• Statement II: xz = 261. Since x and z are integers > 1, x must be a factor of 261. The factors of 261 are 1, 3, 9, 29, 87, 261. Possible values for x (excluding 1) are 3, 9, 29, 87, 261. Among these, 3 and 29 are prime, while 9, 87, and 261 are not. Since x could be prime or not, Statement II alone is not sufficient.

• Since Statement I alone is sufficient and Statement II alone is not, the correct option is (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 because x can be either prime (e.g., 3 or 29) or composite (e.g., 9 or 87).
(c) Select this option if the question can be answered using both the statements together…
The question can be answered using Statement I alone, so it does not require both statements together.
(d) Select this option if the question cannot be answered even using any of the statements
The question can be answered using Statement I alone, so it is not unanswerable.

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.

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