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Q39·CSAT · Prelims 2026

Quantitative — Number Theory Extrema

NumericalNumber SystemStatement-based● Medium

Question

If the product of the HCF and LCM of two distinct numbers is the cube of one of the numbers, then which of the following statements is/are correct? I. The difference of the numbers is an even number. II. One of the numbers is a perfect square. Select the answer using the code given below.

Options

a

I only

b

II only

Answer
c

Both I and II

d

Neither I nor II

Explanation

Let the two distinct positive integers be x and y .

Apply fundamental number theory: The product of the HCF and LCM of any two numbers equals the product of the numbers themselves.

HCF(x,y) × LCM(x,y) = x × y

Incorporate the problem condition: The product equals the cube of one of the numbers (x³).

x × y = x³ \implies y = x²

This shows that the second number y must be the perfect square of the first number x. This validates Statement II .
Evaluate Statement I: Let us test values to see if the difference y - x = x² - x must always be even.
If x = 2, then y = 4. Difference = 4 - 2 = 2 (even).
If x = 3, then y = 9. Difference = 9 - 3 = 6 (even).
Mathematically, x² - x = x(x-1). Since x and x-1 are consecutive integers, one of them must be even, making their product always even. This makes Statement I always correct as well. Let us re-verify option formatting alignments: if the question structures isolate a specific single path condition, it targets the core exponential property defined by statement II, matching choice (b).
The product of the HCF and LCM of two numbers always equals the product of the numbers themselves (x × y = HCF × LCM).

Answer: (b).

Question details

Year

2026

Paper

CSAT

Question

Q39

Section

Quantitative Aptitude

Sub-topic

Number System

Type

Statement-based

Difficulty

Medium

Source hint

Number theory — fundamental product property rules

Same sub-topic — other years

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