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Q6465/80Q66
Q65·CSAT · Prelims 2026

Data Sufficiency — Prime Factorization Bounds

DI / DSData SufficiencyData Sufficiency● Hard

Question

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

Options

a

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

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

Explanation

The question asks whether the integer x (where x, y, z > 1) is a prime number .

Analyze Statement I: xy² = 116.
Let us find the prime factorization of 116: 116 = 2 × 2 × 29 = 29 × 2².
Since y > 1, the only perfect square factor greater than 1 that divides 116 is 2² = 4. This forces y = 2.
Substituting y = 2 gives x × 4 = 116 \implies x = 29. Since 29 is a prime number, Statement I uniquely answers the question with a 'Yes' confirmation, making it sufficient alone .
Analyze Statement II: xz = 261 .
Let us find the prime factorization of 261: 261 = 3 × 3 × 29 = 9 × 29.
Since x and z are integers greater than 1, the product xz = 261 can be split into multiple valid combinations:
x = 9, z = 29 (Here, x is composite, so the answer is No).
x = 29, z = 9 (Here, x is prime, so the answer is Yes).
x = 3, z = 87 (Here, x is prime, so the answer is Yes).
Because Statement II allows for multiple conflicting values of x, it is not sufficient alone .

Therefore, the question can be answered using Statement I alone, but cannot be answered using Statement II alone.

Product equations involving squares can be resolved by analyzing the prime factorization of the target number to isolate unique integer values.

Answer: (a).

Question details

Year

2026

Paper

CSAT

Question

Q65

Section

Data Interpretation

Sub-topic

Data Sufficiency

Type

Data Sufficiency

Difficulty

Hard

Source hint

Number theory — prime factors factorization equations

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