UPSC Prelims 2026 CSAT, question 5

Topic: Data sufficiency

Directions · Q1 to Q5

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 : $xy^2=116$

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 statementCorrect 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

Solution.

Answer: (a) Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement

Statement I alone is sufficient: $x y^2 = 116 = 2^2 \times 29$ ; since $x, y > 1$ are integers, the only square factor greater than $1$ is $y^2 = 4$ ( $y = 2$ ), which uniquely forces $x = 29$ , a prime number.
Statement II alone is not sufficient: $x z = 261 = 3^2 \times 29$ with integers $x, z > 1$ allows $x = 3$ or $x = 29$ (both prime) as well as $x = 9$ or $x = 87$ (both composite).