Class 10Subject Code: 041Year: 202538 Questions Available

CBSE Class 10 Mathematics Standard 2025 — Independently Prepared Solutions

Comprehensive step-by-step solutions for CBSE Class 10 Mathematics Standard 2025 Board Examination (Code 30/1/1), focusing on fundamental algebraic methods, geometric proofs, trigonometric identities, and statistical reasoning.

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Official Source & AttributionSource of question paper: Central Board of Secondary Education (CBSE)

Solutions and conceptual explanations are independently prepared by QuizCluster. QuizCluster is an independent educational platform and is not affiliated with, endorsed by, or sponsored by the Central Board of Secondary Education (CBSE). Official question papers are accessed via direct external links to official CBSE repositories and are never hosted or mirrored on QuizCluster.

Question Index

Q1 of 38

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Fundamental Theorem of Arithmetic (HCF & LCM)

Section A•1 Marks•Chapter: Real Numbers

Easy

Question 1 — Problem Statement (Synopsis)

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Determination of LCM and HCF of two positive integers using prime factor decomposition.

See Question 1 in the official CBSE Question Paper (Set 1)

QuizCluster Step-by-Step Solution

Step 1: Express numbers in prime factorization form

Let the given numbers be a = p³ · q² and b = p · q⁴, where p and q are prime numbers.

Step 2: Apply the product rule for HCF and LCM

HCF(a, b) is the product of the smallest powers of each common prime factor: HCF(a, b) = p¹ · q² = p q². LCM(a, b) is the product of the highest power of each prime factor involved: LCM(a, b) = p³ · q⁴.

Step 3: Calculate the requested value

The question asks for LCM(a, b) / HCF(a, b): (p³ q⁴) / (p q²) = p^(3-1) · q^(4-2) = p² q².

Final Answer

p² q²

Key Formulas & Mathematical Laws

HCF(a, b) = Product of the smallest power of each common prime factor
LCM(a, b) = Product of the greatest power of each prime factor involved
HCF(a, b) × LCM(a, b) = a × b

Important Core Concept

Every composite number can be uniquely expressed as a product of primes, apart from the order in which the prime factors occur.

Common Student Pitfall

Students often swap the power rules: remember HCF takes minimum powers, while LCM takes maximum powers.

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Question 1 of 38

Source of question paper: Central Board of Secondary Education (CBSE). QuizCluster provides independently prepared educational solutions and explanations. QuizCluster is an independent educational platform and is not affiliated with or endorsed by CBSE. Question papers are accessed through links to official CBSE resources and are not hosted by QuizCluster.