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Q63·CSAT · Prelims 2026

Data Sufficiency — Parity Rules

DI / DSData SufficiencyData Sufficiency● Easy

Question

Question : If x and y are integers, then is x even? Statement I : x^2y^2 is even. Statement II : 1+x^2+y^2 is odd.

Options

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

Answer

Explanation

The question asks whether the integer x is definitively even .

Analyze Statement I: x²y² is even.
For a product of squares to be even, at least one of the underlying base variables (x or y) must be even. This leaves a conflict: x could be even (with y odd or even), or x could be odd (if y is even). Since x is not uniquely determined, Statement I is not sufficient alone .
Analyze Statement II: 1 + x² + y² is odd.
Subtracting 1 from both sides implies that x² + y² must be an even number.
For the sum of two squares to be even, both x² and y² must share the same parity: either both are even (meaning both x and y are even) or both are odd (meaning both x and y are odd). Because x can still be either even or odd, Statement II is not sufficient alone.
Combine the Statements: Statement I requires at least one variable to be even, while Statement II requires both variables to share the same parity. Combining these two rules forces both ⟨MATH⟩x⟨/MATH⟩ and ⟨MATH⟩y⟨/MATH⟩ to be even numbers. This resolves the question with a unique 'Yes' confirmation, making the statements sufficient when used together.
Parity equations are solved effectively by analyzing how odd and even components add or multiply to restrict variable possibilities.

Answer: (c).

Question details

Year

2026

Paper

CSAT

Question

Q63

Section

Data Interpretation

Sub-topic

Data Sufficiency

Type

Data Sufficiency

Difficulty

Easy

Source hint

Number theory — integer parity arithmetic

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