IAS 2026 — CSAT Reasoning

2026mcqhard

Directions: 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: $xy^2 = 116$
Statement II: $xz = 261$

Official previous-year question

Verified 2 Aug 2026.

Options

ASelect this option if the question can be answered using one of these statements alone, but cannot be answered using other statement
BSelect this option if the question can be answered using either statement alone
CSelect this option if the question can be answered using both the statements together, but cannot be answered using either statement alone
DSelect this option if the question cannot be answered even using any of the statements

Did you get this right?

Solution

The correct option is (a) - Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement.

[as per provisional answerkey]

Analysis

Statement I alone: $xy^{2} = 116$
We are given that x, y, and z are integers greater than 1. Let's find the prime factorization of 116:
$116 = 2 \times 58 = 2 \times 2 \times 29 = 29 \times 2^{2}$ .
Since the equation is $x \times y^{2} = 116$ and y must be an integer greater than 1, the only possible value for $y^{2}$ is $2^{2}$ (which is 4).
If $y^{2} = 4$ , then $y = 2$ (which is > 1).
Substituting this back: $x \times 4 = 116$ , which gives $x = 29$ .
Since 29 is a prime number, we get a definitive "Yes" to the question "Is x a prime number?".
Statement I alone is sufficient.

Statement II alone: $xz = 261$
We are given $xz = 261$ , where x and z are integers greater than 1. Let's find the factors of 261:
$261 = 3 \times 87 = 3 \times 3 \times 29 = 9 \times 29$ .
Possible pairs for (x, z) such that both are greater than 1 are:
1. $x = 3$ , $z = 87$ (x is prime)
2. $x = 9$ , $z = 29$ (x is NOT prime)
3. $x = 29$ , $z = 9$ (x is prime)
4. $x = 87$ , $z = 3$ (x is NOT prime)
Because x can be either prime (3, 29) or composite (9, 87), we cannot determine if x is definitely a prime number.
Statement II alone is not sufficient.

Why the other options are incorrect

Option (b) - either statement alone: This is incorrect because Statement II fails to provide a unique answer (x could be 9 or 29), whereas Statement I provides a unique prime value for x.
Option (c) - both statements together: This is incorrect because Statement I is already sufficient on its own. The "together" option is only chosen when neither statement works independently.
Option (d) - cannot be answered: This is incorrect because Statement I provides a definitive mathematical proof that x must be 29, which is a prime number.

Key Concept

Prime factorization and integer constraints (x, y > 1) in algebraic equations to determine unique values.

Your note

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Options

ASelect this option if the question can be answered using one of these statements alone, but cannot be answered using other statement

BSelect this option if the question can be answered using either statement alone

CSelect this option if the question can be answered using both the statements together, but cannot be answered using either statement alone

DSelect this option if the question cannot be answered even using any of the statements

Solution

The correct option is (a) - Select this option if the question can be answered using one of these statements alone, but cannot be answered using other statement.

[as per provisional answerkey]

Analysis

Why the other options are incorrect

Option (b) - either statement alone: This is incorrect because Statement II fails to provide a unique answer (x could be 9 or 29), whereas Statement I provides a unique prime value for x.
Option (c) - both statements together: This is incorrect because Statement I is already sufficient on its own. The "together" option is only chosen when neither statement works independently.
Option (d) - cannot be answered: This is incorrect because Statement I provides a definitive mathematical proof that x must be 29, which is a prime number.

Key Concept

Prime factorization and integer constraints (x, y > 1) in algebraic equations to determine unique values.

IAS 2026 — CSAT Reasoning

Options

Did you get this right?

Solution

Analysis

Why the other options are incorrect

Key Concept

Your note

IAS Reasoning in other years

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Other CSAT topics for IAS

Comprehension

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Keep practicing Reasoning

IAS 2026 — CSAT Reasoning

Options

Did you get this right?

Solution

Analysis

Why the other options are incorrect

Key Concept

Your note

IAS Reasoning in other years

More IAS Reasoning questions

Other CSAT topics for IAS

Comprehension

Maths

Keep practicing Reasoning