NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Class 10 Maths Chapter 1 Solutions
Class 10 Maths Chapter 1 Main Topics
Class 10 Maths Chapter 1 Practice Tests
10th Maths Chapter 1 Question wise Solutions
Class 10 Maths Chapter 1 MCQ
Class 10 Maths Chapter 1 Case Study
Class 10 Maths Chapter 1 Assignments

1. Show that 6ⁿ cannot end with digit 0 for any natural number n. [CBSE 2023]
See SolutionFor a number to end in 0, it must be divisible by 10.
10 = 2 × 5
Prime factorization of 6 = 2 × 3
Any power of 6ⁿ will only have 2 and 3 as prime factors.
Hence, it can never be divisible by 5.
Therefore, 6ⁿ cannot end in 0.

2. Find the HCF and LCM of 72 and 120. [CBSE 2023]
See SolutionPrime factorization:
72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF = 2³ × 3 = 24
LCM = 2³ × 3² × 5 = 360

3. Using prime factorisation, find HCF and LCM of 96 and 120. [CBSE 2023]
See Solution96 = 2⁵ × 3
120 = 2³ × 3 × 5
HCF = 2³ × 3 = 24
LCM = 2⁵ × 3 × 5 = 480

4. Find the greatest 3-digit number which is divisible by 18, 24 and 36. [CBSE 2023]
See SolutionFind LCM of 18, 24, 36
= 2³ × 3² = 72
Greatest 3-digit number = 999
Divide: 999 ÷ 72 = 13.875
→ Take floor = 13
13 × 72 = 936

5. If √2 is an irrational number, prove (5 – 2√2) is also an irrational number. [CBSE 2023 Compt]
See SolutionAssume 5 – 2√2 is rational.
Then, 2√2 = 5 – (rational number) = rational
⇒ √2 = rational
⇒ Contradiction.
So, 5 – 2√2 is an irrational number.

6. Prove that √2 is an irrational number. [CBSE 2023, 2020 Compt]
See SolutionLet √2 = a/b in lowest terms.
⇒ 2 = a²/b²
⇒ a² = 2b²
⇒ a is even
⇒ a = 2k
⇒ (2k)² = 4k² = 2b²
⇒ b² = 2k²
⇒ b is even
⇒ Both a and b even
⇒ Contradiction.
So, √2 is irrational.

7. If p = ab³ and q = a²b, a and b are prime, find LCM(p, q). [CBSE 2020 Compt]
See Solutionp = ab³
q = a²b
LCM = a²b³

8. Prove that √3 is an irrational number. [CBSE 2020]
See SolutionSame method as √2 proof.
Assume √3 = a/b
⇒ 3 = a²/b²
⇒ a² = 3b²
⇒ a is divisible by 3, let a = 3k
⇒ (3k)² = 9k² = 3b²
⇒ b² = 3k²
⇒ b divisible by 3
⇒ Both divisible by 3
⇒ Contradiction
So, √3 is an irrational number.

9. The traffic lights at three different road crossings change after every 48s, 72s, and 138s. If they change simultaneously at 7am, at what time will they change together next? [CBSE 2020]
See SolutionLCM(48, 72, 138) = LCM = 432 sec
Therefore, after 432 s, they will change simultaneously. We know that 60s = 1 min
= 1 s = 1/60 min
= 432 s = 432/60 min = 7 min 12 s
Hence, the lights will change simultaneously at 7 : 07 : 12 am.

10. Find the HCF and LCM of 26, 65, 117 using prime factorisation. [CBSE 2020]
See Solution26 = 2 × 13
65 = 5 × 13
117 = 3 × 3 × 13
HCF = 13
LCM = 2 × 3² × 5 × 13 = 1170.

11. Prove that √5 is an irrational number. [CBSE 2020]
See SolutionSame process as √2 proof.
Let √5 = a/b
⇒ 5 = a²/b²
⇒ a² = 5b²
Then a and b both divisible by 5
⇒ contradiction
⇒ √5 is irrational number.

12. Prove 2 + 5√3 is an irrational number, given √3 is irrational. [CBSE 2019]
See SolutionAssume 2 + 5√3 is rational
⇒ Subtract 2: 5√3 = rational
⇒ √3 = rational, which is contradiction.
So, 2 + 5√3 is irrational.

13. Prove (2 + √3)/5 is an irrational, given √3 is an irrational number. [CBSE 2019]
See SolutionAssume (2 + √3)/5 is rational
⇒ 2 + √3 = rational
⇒ √3 = rational
⇒ Contradiction
So, (2 + √3)/5 is irrational.

14. Write the smallest number divisible by both 306 and 657. [CBSE 2019]
See SolutionFind LCM(306, 657)
Prime factors:
306 = 2 × 3² × 17
657 = 3 × 219 = 3 × 3 × 73 = 3² × 73
LCM = 2 × 3² × 17 × 73 = 22338

15. a = x³y² and b = xy³, where x and y are primes. Find LCM(a, b). [CBSE 2019]
See SolutionGiven two positive integers
a = x³y² = x × x × x × y × y 
and b = xy³ = x × y × y × y
LCM (a,b) = Product of the greatest power of each prime factor involved in the numbers
 = x³ × y³ 
LCM = x³ y³

16. If HCF(336, 54) = 6, find LCM(336, 54). [CBSE 2019]
See SolutionProduct of numbers = HCF × LCM
⇒ 336 × 54 = 6 × LCM
⇒ LCM = (336 × 54)/6 = 3024

17. LCM is 9 times HCF. LCM + HCF = 500. Find HCF.  [CBSE 2019 Compt]
See SolutionLet HCF = x ⇒ LCM = 9x
x + 9x = 500
⇒ 10x = 500
⇒ x = 50

18. Show that (3 + √7)/5 is an irrational number, given √7 is irrational. [CBSE 2019 Compt]
See SolutionAssume (3 + √7)/5 is rational
⇒ Multiply both sides by 5: 3 + √7 = rational
⇒ √7 = rational
⇒ Contradiction
So, (3 + √7)/5 is irrational

Textbook solutions, available in PDF formats are useful for grasping the intricacies of the syllabus. To reinforce learning, practice problems and MCQs from chapter 1 are indispensable. NCERT Class 10 Maths Book Chapter 1 Solutions PDF Download provides an easy way for students to access these resources on-the-go, making revision and exam preparation more efficient. Class 10 Mathematics textbook explanation guide of chapter 1 is prepared according to books published for 2025-26 CBSE textbooks Syllabus.

Preparing for mathematics exams becomes simpler with NCERT Class 10 Maths Chapter 1 Important Questions and CBSE Previous Year Questions. These resources help students focus on frequently asked problems, enhancing their problem-solving skills. For instance, understanding Real Numbers in depth requires familiarity with proofs of irrational numbers and applications of the Fundamental Theorem of Arithmetic. Textbook Class 10 Maths Chapter 1 Notes and Summary act as quick references, saving time while revising key concepts.

Solving 10th Math Exercise 1.1 to Exercise 1.4 further aids in solidifying these concepts. For students looking for additional support, online video lectures and study materials provide step-by-step explanations of the chapter’s topics. Incorporating NCERT Complete Solutions for Real Numbers Class 10 into daily study routines can significantly boost confidence, ensuring students are well-prepared to perform in their board exams. Learning Class 10 Maths Chapter 1 Real Numbers effectively requires a systematic approach and a clear understanding of the concepts involved. Here are some steps and strategies to help you learn class 10 mathematics chapter 1.

Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that: a = bq + r, where 0 ≤ r < b.
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of factors.

Key Highlights of Class 10 Maths Chapter 1 Real Numbers

Class 10 Maths Chapter 1, Real Numbers, focuses on Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, proving the irrationality of numbers, terminating and non-terminating decimals and revisiting HCF and LCM. These concepts form the basis of board exam questions, often involving direct proofs, applications and numerical problems. Mastery ensures strong foundational skills.

Class 10 Maths NCERT Book Chapter 1 Real Numbers is not only useful for CBSE board exams but also lays the groundwork for advanced mathematics. Tools like NCERT Class 10 Maths Chapter 1 Exercises Formula List and Practice Problems simplify understanding and application of theorems. Teachers often recommend using NCERT Simplified Solutions for Real Numbers Class 10 to enhance topics clarity. Students can also test their knowledge with online tests, MCQs and sample papers prepared for Chapter 1. For comprehensive preparation, the Class 10 Maths Chapter 1 Exercise 1.1, Exercise 1.2 and Exercise 1.3 Solutions are particularly helpful in mastering the topic step by step. Resources such as Class 10th NCERT Maths Textbook Chapter 1 Study Material ensure that learners have everything they need in one place. With consistent practice and a focus on important questions, students can confidently approach any problem from this critical chapter.

Methods for proving the irrationality of certain numbers, such as the square root of a prime number, are discussed. Understanding decimal expansions of real numbers. Terminating decimals and recurring decimals. Rationalization is the process of converting an irrational number into a rational one by multiplying or dividing by a suitable number.

According to new syllabus for academic session, only two exercises are in chapter 1. The exercise 1.1 deleted and exercise 1.2 become the exercise 1.1. Similarly exercise 1.3 become exercise 1.2 and exercises 1.4 is also deleted.
The course structure of class 10 mathematics chapter 1 is given below:
Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and
after illustrating and motivating through examples, Proofs of irrationality of √2, √3, √5.

>> Total Number of Exercises: 2
>> Total Marks in Board Exam: 6
>> Number of Periods needed: 15
Class 10 Maths Chapter 1 Real Numbers, is an important chapter that lays the foundation for understanding various mathematical concepts. The main points covered in this chapter include introduction to Real Numbers. Real numbers are a set of numbers that includes all rational and irrational numbers. Real numbers can be represented on the number line.

Utilize online resources like video tutorials, educational websites like Tiwari Academy and interactive quizzes to reinforce your understanding of the chapter. As you progress, practice solving sample papers or past year question papers. This will help you become familiar with the exam pattern and time management. Periodically test your knowledge by attempting self-assessment quizzes or problems from the chapter. This will help you gauge your progress.

In class 10 Maths Chapter we will learn about understanding square numbers and their properties. Finding the square root of a number using prime factorization. Real numbers are used in various mathematical applications, such as finding the distance between two points in coordinate geometry.

Class 10 Maths Chapter Real Numbers
The chapter 1 provides numerous examples and practice problems to reinforce the concepts discussed. These are the main points covered in Class 10 Maths Chapter 1 Real Numbers. It serves as the building block for further mathematical concepts and is essential for a solid foundation in mathematics. Get here topic wise solutions for 10th Mathematics chapter 1 all exercises in Hindi and English medium. Students can take help of videos for solving questions of each exercise.

Class 10 Maths Chapter 1 Exercise 1.1 Question 1

Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
The step by step solutions of each part is given here with practice questions>. You can discuss with us Part (i), Part (ii), Part (iii), Part (iv), and Part (v) to learn more about the solution.

Class 10 Maths Chapter 1 Exercise 1.1 Question 2
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54.
Get the solution of question 2 and discuss Part (i), Part (ii) and Part (iii) with us along with assignments and other doubts for more clarification.

Class 10 Maths Chapter 1 Exercise 1.1 Question 3
Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25.
Discuss with part one, two and three in detail and get worksheets to understand the solution in simple way.

Class 10 Maths Chapter 1 Exercise 1.1 Question 4
Given that HCF (306, 657) = 9, find LCM (306, 657).
The solution of this question is very simple and usually asked in MCQ type questions. For more revision, please write to us.

Class 10 Maths Chapter 1 Exercise 1.1 Question 5
Check whether 6ⁿ can end with the digit 0 for any natural number n.
The solution of question 5 is similar to the example given in textbook. Discuss with us and Practice here more questions based on question 5.

Class 10 Maths Chapter 1 Exercise 1.1 Question 6
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Extra questions for practice based on the solution of question 6 are given here to download. Put your view to discuss with us.

Class 10 Maths Chapter 1 Exercise 1.1 Question 7
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
Question number 7 is important for exams. Get here the solution and practice assignments with answers for revision.

Class 10 Maths Chapter 1 Exercise 1.2 Question 1
Prove that √5 is irrational.
To understand the solution of question 1, student need to practice assignments and discuss the explanation in depth.

Class 10 Maths Chapter 1 Exercise 1.2 Question 2
Prove that 3 + 2√5 is irrational.
Get here the assignments and practice questions based on the solution of question 2. If still there is any doubt, please visit to discussion forum.

Class 10 Maths Chapter 1 Exercise 1.2 Question 3
Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2
The solution of question 3 is similar to question 1 and 2. Get more practice to learn the concepts in better way. Discuss in detail here.

Use number lines and diagrams to visualize concepts like decimal expansions and the placement of real numbers on the number line. Visual aids can help make abstract concepts more concrete. Don’t hesitate to ask your teacher or a classmate for help if you encounter difficulties. Sometimes, discussing problems with others can provide valuable insights. Maintain a study schedule and allocate specific time slots for learning and practicing this chapter.

The complete NCERT chapter 1 of 10th Maths 1 is divided in four main parts. The explanation of each part with video explanation is given below. Start by reading the chapter thoroughly and grasp the basic definitions and concepts. Ensure that you understand the definitions of real numbers, Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic. Take concise notes while reading the chapter. Write down key formulas, definitions and important points. These notes will be helpful for quick revision.

Learn here how to find the prime factorization of numbers. This skill is essential for various aspects of number theory. Practice finding the prime factorization of different numbers. Practice working with terminating and recurring decimals. Solve NCERT textbook problems that involve decimal expansions of real numbers. Familiarize yourself with methods for rationalizing irrational numbers, especially square roots. Practice these techniques with different examples. Remember that learning mathematics often requires patience and persistence. Take your time to understand each concept thoroughly before moving on to the next. By following these steps and strategies, you can effectively learn Class 10 Maths NCERT Chapter 1 Real Numbers.

Class 10 Maths Chapter 1 Main Topics
Mathematics is best learned through practice. Solve a variety of problems related to real numbers, including examples from your textbook and additional practice questions. Pay attention to different types of questions, such as finding HCF, LCM, prime factorization and working with decimal expansions.

Euclid’s Division Lemma is a fundamental concept in this chapter. Make sure you understand how it works and can apply it to solve problems related to divisibility and remainders. There are four exercises in chapter 1 covering all the four topics on Real Numbers. Students can practice well using the 6 question papers given below. It will help to prepare the chapter for CBSE exams also.

The word algorithm come from the name of the name of the 9th century Persian mathematician al-Khwarizmi. The word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala.

An equivalent version of Fundamental theorem of Arithmetic was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic.

However, the first correct proof was given by Carl Friedrich Gauss in his al-Khwarizmi.
Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians’ and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science. Learn here how to make NCERT class 10 Maths chapter 1 easy to learn.