NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (original) (raw)
Last Updated : 23 Jul, 2025
**NCERT Solutions for Class 10 Maths Chapter 1 **Real Numbers is an essential tool in the form of an article created by our team at GfG to help students in their academic journey. The team at GFG is a well-wisher of students and always prioritizes the benefit of learners by creating beneficial content in the form of articles. This article covers NCERT Class 10 Maths Solutions for Chapter 1, which is about Real Numbers.
The NCERT Solutions for Class 10 Maths Chapter 1: Real Numbers is a crucial resource for students seeking to master fundamental concepts in mathematics. Created by the dedicated team at GfG, this article aims to support students in understanding and applying key topics covered in this chapter. By providing detailed explanations and solutions, we strive to enhance students' grasp of Real Numbers and prepare them effectively for their exams.
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Topics:
This section covers the essential topics from Chapter 1, including Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, and methods for finding the HCF and LCM, all crucial for mastering real numbers.
The following list contains all the topics covered in Chapter 1 of Class 10 Maths Real Numbers:
| Section Name | Topic Name |
|---|---|
| 1.1 | Introduction |
| 1.2 | Euclid’s Division Lemma |
| 1.3 | The Fundamental Theorem of Arithmetic |
| 1.4 | Revisiting Irrational Numbers |
| 1.5 | Revisiting Rational Numbers and Their Decimal Expansions |
| 1.6 | Summary |
Euclid's Division Lemma
Here, we explore Euclid’s Division Lemma, a fundamental principle in mathematics that forms the basis for understanding division in number theory.
- The Division Lemma states that given two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
- In simpler terms, the Division Lemma says that any positive integer can be divided by another positive integer, and the result will be a quotient and a remainder.
- This concept is important in many areas of mathematics, including number theory, algebra, and geometry.
Euclid's Division Algorithm
- Euclid's Division Algorithm is used for finding the HCF of two numbers. Let's say the number is a and b such that a>b.
- Now, By using the Euclid Divison Lemma we find two Integer p and r such that a=b×q+r and 0≤r<b.
- If r = 0, the H.C.F is b; else, we apply Euclid’s division Lemma to b (the divisor) and r (the remainder) to get another pair of quotients and remainder.
- We repeat the above Step until a remainder of zero is obtained. The divisor in that step is the H.C.F. of the given set of numbers.
Fundamental Theorem of Arithmetic
This part discusses the Fundamental Theorem of Arithmetic, which emphasizes the unique prime factorization of numbers, an essential concept in mathematics.
The fundamental theorem of Arithmetic states that the multiplication of prime factorization of every number is unique if we ignore the arrangement of prime factors of a given number.
Example:- Prime Factors of 1092 are 2,3,2,7,13 or 2,7,2,3,13
Therefore, 1092 is represented as a product of prime factors (Two 2s, 3, 7 and 13) ignoring the arrangement of the factors.
Methods of Finding HCF
HCF is the greatest number that divides each of the given numbers without leaving any remainder is the highest common factor (HCF) of two or more given numbers
Example:- Here are the steps to find the HCF of 12 and 15:
- **Step 1: List the factors of 12: 1, 2, 3, 4, 6, 12
- **Step 2: List the factors of 15: 1, 3, 5, 15
- **Step 3: Identify the common factors: 1 and 3
- **Step 4: The highest common factor is the greatest common factor among the common factors, which in this case is 3. So, the **HCF of 12 and 15 is 3.
Methods of Finding LCM
LCM is the smallest of the common multiples of two or more numbers and is called the lowest common multiple (LCM).
Example:- LCM of 510 and 92
- **Step 1: 510 = 2 x 3 x 5 x 17
- **Step 2: 92 = 2 x 2 x 23
- **Step 3: Common factors are 2
- **Step 4: uncommon factors are 3x2, 5x23 and 17
- **Step 5: Then, we take the highest power of each prime factor that appears in either number and multiply them together: 2^2 x 3 x 5 x 17 x 23 = 23,460. So, the LCM of 510 and 92 is 23,460.
Application of LCM and HCF in Real World Problems
- LCM can be used to find the point of common occurrences. for example:- finding the time at which two persons are encountered running at distinct speeds.
- HCF can be used in computer science and cryptography to help with encryption and decryption algorithms.
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercises:
- NCERT Maths Solutions Class 10 Exercise 1.1 – 5 Questions (4 Long Answers, 1 Short Answer)
- NCERT Maths Solutions Class 10 Exercise 1.2 – 7 Questions (4 Long Answers, 3 Short Answers)
- NCERT Maths Solutions Class 10 Exercise 1.3 – 3 Questions (3 Short Answers)
- NCERT Maths Solutions Class 10 Exercise 1.4 – 3 Questions (3 Short Answers)
_Here, you can also find all of the solutions for the _NCERT exercises for CBSE Class 10 Maths.
Key Features of NCERT Solutions for Class 10 Maths Chapter 1- Real Numbers
- These NCERT Solutions are Created by the experienced GfG team putting student's benefit first.
- These Solutions provided here are 100% correct and students can use these solutions for the preparation for their board examination.
- All the solutions provided here are in Step by Step manner with detailed explanations for the steps in between.