NCERT Class 10 Maths Chapter 1: Real Numbers - Solutions for Exercises 1.2 to 1.4
Class 10 Maths Real Numbers – Complete Guide with Step-by-Step Solutions
Master Euclid’s Lemma, Prime Factorization, Irrational Proofs & Exam Tricks in One Place
📑 Table of Contents
Introduction
Mathematics in Class 10 plays a vital role in shaping your problem-solving mindset. Among all chapters, Real Numbers forms the foundation of higher mathematics and is one of the most scoring topics in board exams. This article is designed not only to provide you with answers but also with detailed, step-by-step explanations that help you truly understand the “why” behind every solution.
You will explore Euclid’s Division Lemma, Prime Factorization, Irrational Proofs, and Terminating/Non-Terminating Decimals through practical solved questions. Each solution is broken into easy steps with clear reasoning. In addition, a Bonus Masterstroke Section at the end will reveal rarely taught tips and exam hacks that can help you write answers like a topper.
Exercise 1.2 – Euclid’s Division Lemma
Euclid’s Division Lemma is one of the foundational tools in mathematics for finding the HCF of two numbers. The lemma states that for any two positive integers a and b, there exist integers q (quotient) and r (remainder) such that:
a = bq + r, where 0 ≤ r < b
Q.1 Find the HCF of 225 and 135 using Euclid’s Division Lemma.
Step 1: Divide 225 by 135.
225 = 135 × 1 + 90 (remainder = 90)
Step 2: Now divide 135 by 90.
135 = 90 × 1 + 45 (remainder = 45)
Step 3: Now divide 90 by 45.
90 = 45 × 2 + 0 (remainder = 0)
Hence, HCF = 45 ✅
Q.2 Prove that √3 is an irrational number.
Step 1: Assume √3 is rational. Then it can be expressed as p/q, where p and q are integers with no common factors.
Step 2: Squaring both sides, 3 = p² / q² ⇒ p² = 3q².
Step 3: This implies p² is divisible by 3, so p is divisible by 3.
Step 4: Let p = 3k. Then p² = 9k² = 3q² ⇒ q² = 3k². This means q is also divisible by 3.
Step 5: But this contradicts the assumption that p and q have no common factor.
Hence, √3 is irrational ✅
Exercise 1.3 – Fundamental Theorem of Arithmetic
This theorem states that every composite number can be expressed as a product of primes, and this factorization is unique apart from the order of primes.
Q.1 Express 140 as a product of prime factors.
Step 1: Divide 140 by 2 → 70.
Step 2: Divide 70 by 2 → 35.
Step 3: Divide 35 by 5 → 7.
Step 4: 7 is prime.
Therefore, 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7 ✅
Q.2 Find LCM and HCF of 90 and 144 using prime factorization.
90 = 2 × 3² × 5
144 = 2⁴ × 3²
HCF: Take the lowest power of common primes = 2 × 3² = 18
LCM: Take the highest power of all primes = 2⁴ × 3² × 5 = 720
Answer: HCF = 18, LCM = 720 ✅
Exercise 1.4 – Rational Numbers and Decimals
This exercise deals with proving whether decimals are terminating or non-terminating repeating using prime factorization of denominators.
Q.1 Show that 1/8 has a terminating decimal expansion.
Denominator 8 = 2³ (only prime factor 2).
Hence it terminates.
1/8 = 0.125 ✅
Q.2 Show that 1/7 has a non-terminating repeating decimal expansion.
Denominator 7 is prime and not 2 or 5.
Hence decimal expansion repeats forever.
1/7 = 0.142857142857… (repeats) ✅
💡 Bonus Masterstroke Knowledge
- Shortcut: If denominator (in simplified form) has prime factors other than 2 or 5 → non-terminating repeating decimal.
- Exam hack: Always write “By Fundamental Theorem of Arithmetic…” before prime factorization to earn method marks.
- Real life: LCM is used for traffic signal timings; HCF is used in arranging groups or batches.
- Memory trick: For irrational proofs, always assume rational → contradiction → hence irrational.
FAQs
Q1. What is the easiest way to find HCF?
Use Euclid’s Division Lemma—it’s faster and examiners prefer it for proofs.
Q2. How do I check if a decimal will terminate?
Simplify the fraction. If denominator has only 2s or 5s → terminating. Otherwise repeating.
Q3. Is memorizing prime factorization tables necessary?
Not at all. Practicing division methods is enough; memorization is optional.
Q4. Why are irrational numbers important?
They are used in advanced mathematics, geometry (like diagonal of a square), and modern technology such as encryption algorithms.
About the Author
Zayyan Kaseer is a passionate educator dedicated to simplifying mathematics for students. With years of teaching experience, Zayyan blends deep subject knowledge with warm, relatable examples that make learning enjoyable and practical.
Disclaimer
This article is for educational purposes only. While utmost care has been taken to ensure accuracy, always cross-check with official NCERT textbooks or consult your teacher for clarification.
Closing Motivational Message
Dear students, remember: Mathematics is not about rote learning—it is about patterns, logic, and discovery. Every problem solved is a victory, every mistake is a step closer to mastery. Keep practicing with confidence—you are capable of far more than you believe! 🚀
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