NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers

This worksheet provides complete NCERT Solutions for Class 10 Mathematics Chapter Real Numbers. This chapter introduces students to important concepts related to numbers, such as factors, multiples, highest common factor (HCF), least common multiple (LCM), and properties of irrational numbers. It helps students understand how numbers are connected and how mathematical reasoning works in solving real-life problems. The worksheet provides complete and accurate NCERT Solutions that follow the official NCERT pattern, helping students build confidence and prepare effectively for exams

Chapter summary: stories, poems & themes

This chapter focuses on mathematical concepts related to numbers and their properties. It includes problem-solving tasks and logical reasoning activities based on factorisation, HCF, LCM, and irrational numbers. Students learn how to express numbers as products of primes, verify mathematical relationships, and understand proofs using contradiction. The chapter is concept-based and activity-driven, helping students develop strong mathematical thinking and reasoning skills.

What this NCERT chapter covers?

• Prime factorisation of numbers 
• Finding HCF and LCM of numbers 
• Verification of mathematical relationships 
• Understanding rational and irrational numbers 
• Logical reasoning and proof using contradiction 
• Problem-solving and numerical calculations 

How to use these NCERT solutions?

Students should first read each question carefully and attempt to solve it independently before checking the answer. Parents and teachers can guide students in understanding the correct method used in the solutions. These solutions follow the exact NCERT order and structure, making them useful for revision, homework support, and exam preparation. Practising regularly with these solutions helps students understand concepts clearly and improve accuracy.

Student tips & learning tricks

• Always write numbers in prime factor form before finding HCF or LCM 
• Check calculations carefully while multiplying factors 
• Follow each step clearly when solving proof-based questions 
• Revise formulas and definitions regularly 
• Read the question carefully to avoid calculation mistakes 

Why NCERT solutions are important?

NCERT Solutions help students understand mathematical concepts in a structured and reliable way. They build a strong foundation in mathematics and improve problem-solving skills. Using NCERT-aligned solutions ensures that students learn correct methods and stay prepared for school exams and assessments. Regular practice with accurate solutions increases confidence and helps students perform better academically.

Complete answer key – NCERT solutions

Exercise 1.1

1. (i) 140 = 2 × 2 × 5 × 7 
(ii) 156 = 2 × 2 × 3 × 13 
(iii) 3825 = 3 × 3 × 5 × 5 × 17 
(iv) 5005 = 5 × 7 × 11 × 13 
(v) 7429 = 17 × 19 × 23 

2. (i) 26 = 2 × 13, 91 = 7 × 13 
HCF = 13 
LCM = 2 × 7 × 13 = 182 
Verification: 13 × 182 = 26 × 91 

(ii) 510 = 2 × 3 × 5 × 17, 92 = 2 × 2 × 23 
HCF = 2 
LCM = 2 × 2 × 3 × 5 × 17 × 23 = 23460 
Verification: 2 × 23460 = 510 × 92 

(iii) 336 = 2⁴ × 3 × 7, 54 = 2 × 3³ 
HCF = 2 × 3 = 6 
LCM = 2⁴ × 3³ × 7 = 3024 
Verification: 6 × 3024 = 336 × 54 

3. (i) 12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7 
HCF = 3 
LCM = 2² × 3 × 5 × 7 = 420 

(ii) 17, 23, 29 are primes 
HCF = 1 
LCM = 17 × 23 × 29 = 11339 

(iii) 8 = 2³, 9 = 3², 25 = 5² 
HCF = 1 
LCM = 2³ × 3² × 5² = 1800 

4. LCM = (306 × 657) ÷ 9 
= (306 ÷ 9) × 657 
= 34 × 657 
= 22338 

5. 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ 
Since it has no factor 5, it cannot end with 0. 

6. 7 × 11 × 13 + 13 
= 13(7 × 11 + 1) 
= 13 × 78 
= composite 

7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 
= 5(7 × 6 × 4 × 3 × 2 × 1 + 1) 
= composite 

7. Time taken = LCM of 18 and 12 
18 = 2 × 3², 12 = 2² × 3 
LCM = 2² × 3² = 36 minutes 

Exercise 1.2

1.  Assume √5 is rational. 
So, √5 = a/b (a, b coprime) 
⇒ 5b² = a² 
⇒ 5 divides a² ⇒ 5 divides a 
Let a = 5c 
⇒ 5b² = 25c² 
⇒ b² = 5c² ⇒ 5 divides b 

Contradiction (a and b not coprime) 
∴ √5 is irrational 

2. Assume 3√2 + 5 is rational 
⇒ 3√2 = rational number 
⇒ √2 is rational 
Contradiction 
∴ 3√2 + 5 is irrational 

3. (i) Assume 1/√2 is rational 
⇒ √2 is rational 
Contradiction 
∴ 1/√2 is irrational 

(ii) Assume 7√5 is rational 
⇒ √5 is rational 
Contradiction 
∴ 7√5 is irrational 

(iii) Assume 6 + √2 is rational 
⇒ √2 is rational 
Contradiction 
∴ 6 + √2 is irrational Get structured learning support and step-by-step guidance to strengthen mathematical understanding and improve exam performance.

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