UPSC CSE 2026 · PYQ · Data Sufficiency / Number Theory · medium
Directions: Each item in this section contains a question followed by two statements. Answer each item using the following instructions:
(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.(a)
B.(b)✓ Correct
C.(c)
D.(d)
Explanation
Statement I: xy² = 116 = 4 × 29 = 2² × 29. With y > 1, y² is a perfect square > 1. Possible: y²=4, x=29 (prime). y²=29 not integer. y²=116/x: only y=2, x=29 works. So x=29, prime. Sufficient. Statement II: xz = 261 = 9 × 29 = 3² × 29. With x, z > 1: (x,z) could be (3,87), (9,29), (29,9), (87,3), (261/... ). x=3 prime, x=9 not prime, x=29 prime, x=87=3×29 not prime. Multiple values—not sufficient by itself. Wait, both x=3 (prime) and x=9 (not prime) possible, so Statement II alone insufficient. Statement I alone gives x=29 uniquely (prime). So answer is (a)—only Statement I suffices. Correction: option (a) per the instruction means one alone works but not the other. Hence (a).
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