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# NCERT Solutions for Class 10 Maths Chapter 1 : Real Numbers

Class 10 maths chapter 1 Real Numbers deals with the foundation of numeric system. Students preparing for their Class 10 exams will be able to clear all their concepts on real numbers class 10 at the root level. Expert faculty of Toppr produced these solutions to assist students with their first term exam preparations. It covers all of the major concepts in detail, allowing students to understand the ideas better.

NCERT solutions for class 10 maths chapter 1 concentrates on the essential concepts such as Euclid's division lemma, Prime Numbers, Composite Numbers, Fundamental Theorem of Arithmetic, HCF and LCM by Prime Factorization Method, and Irrational Numbers, etc.

Maths class 10 chapter 1 Questions and Answers are very useful for getting good grades in tests and properly preparing you with all of the important concepts. These NCERT Solutions are valuable tools that can assist you not only in covering the full syllabus but also in providing in-depth analysis of the subjects. The Class 10 Maths NCERT Solutions Chapter 1 are available in pdf format below, and some of them are also included in the exercises.

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## Access NCERT Solutions for Class 10 Maths Chapter 1 : Real Numbers

Exercise 1.1
Question 1
Use Euclid's division algorithm to find the HCF of:
(i) and (ii) and (iii) and
Find the highest HCF among them.
Medium
Solution Verified by Toppr

(i) By Euclid's division lemma,
So,H.C.F of and is

(ii) By Euclid's division lemma,
So, H.C.F of and is

(iii) By Euclid's division lemma,
So, H.C.F of and is

The highest HCF among the three is .
Question 2
Show that any positive odd integer is of the form , or , or , where is some integer.
Hard
Solution Verified by Toppr
Using Euclid division algorithm, we know that ----(1)

Let be any positive integer and .

Then, by Euclid’s algorithm, for some integer , and , .

Therefore,

is divisible by , so it is an even number.

is divisible by , but is not divisible by so it is an odd number.

is divisible by , and is divisible by so it is an even number.

is divisible by , but is not divisible by so it is an odd number.

is divisible by , and is divisible by so it is an even number.

is divisible by , but is not divisible by so it is an odd number.

And therefore, any odd integer can be expressed in the form
Question 3
An army contingent of members is to march behind an army band of members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Hard
Solution Verified by Toppr
HCF () is the maximum number of columns in which they can march.

Step 1: First find which integer is larger.

Step 2: Then apply the Euclid's division algorithm to and to obtain

Repeat the above step until you will get remainder as zero.

Step 3: Now consider the divisor 32 and the remainder 8, and apply the division lemma to get

Since the remainder is zero, we cannot proceed further.

Step 4: Hence the divisor at the last process is

So, the H.C.F. of and is

Therefore, is the maximum number of columns in which they can march.
Question 4
Use Euclid's division lemma to show that the square of any positive integer is either of the form or for some integer m, but not of the form
Hard
Solution Verified by Toppr
Let be the positive integer and .

We know ,

Now, ,

The possibilities of remainder is or .

Case 1 : When

where

Case 2 : When

where

Case 3: When

where

Hence, from all the above cases, it is clear that square of any positive integer is of the form or .
Question 5
Use Euclid's division lemma to show that the cube of any positive integer is of the form or
Hard
Solution Verified by Toppr

Let be any positive integer. Then, it is of the form 3or, 3+ 1 or, 3+ 2.

So, we have the following cases :

Case I : When = 3q.

then, x3 = (3q)3 = 27q3 = 9 (3q3) = 9m, where = 3q3.

Case II : When = 3+ 1

then, x3 = (3+ 1)3

= 27q3 + 27q2 + 9+ 1

= 9 (3q2 + 3+ 1) + 1

= 9+ 1, where (3q2 + 3+ 1)

Case III. When = 3+ 2

then, x3 = (3+ 2)3

= 27 q3 + 54q2 + 36+ 8

= 9(3q2 + 6+ 4) + 8

= 9 + 8, where (3q2 + 6+ 4)

Hence, xis either of the form 9 or 9 + 1 or, 9 + 8.

Exercise 1.2
Question 1
Express each number as a product of its prime factors:
(i) (ii) (iii) (iv) (v)
Medium
Solution Verified by Toppr
(i)

(ii)

(iii)

(iv)

(v)
Question 2
Find the LCM and HCF of the following pairs of integers and verify that LCM HCF Product of the integers:
(i) and (ii) and (iii) and
Easy
Solution Verified by Toppr
1) and
and
verification:

2) and
and
verification:

3) and
and
verification:
Question 3
Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) and (ii) and (iii) and
Medium
Solution Verified by Toppr
Using prime factorisation method:

(i) 12, 15 and 21

Factor of

Factor of

Factor of

HCF

LCM

(ii) 17, 23 and 29

Factor of

Factor of

Factor of

HCF

LCM

(iii) 8, 9 and 25

Factor of

Factor of

Factor of

HCF

LCM
Question 4
Given that HCF and , find LCM of and
Medium
Solution Verified by Toppr
We know that,

Product of the two numbers

LCM

LCM
Question 5
Check whether can end with the digit for any natural number .
Medium
Solution Verified by Toppr
If any digit has the last digit that means it divisible by

The factor of

So value of should be divisible by and

Both is divisible by but not divisible by

So, it can not end with
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### NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers – Brief Overview

Both rational and irrational numbers are included in real numbers. A rational number is a number that can be stated as a fraction of two integers, with the denominator not equal to zero. Irrational numbers are those that cannot be stated as fractions. Irrational numbers have decimal expansions that don't end and don't become periodic. Integers, decimals, and fractions are examples of rational numbers, whereas irrational numbers include root overs, pi (22/7), and so on. In a nutshell, real numbers are all numbers other than imaginary numbers.

#### Euclid’s Division Lemma

Euclid's division algorithm is a method for calculating the Highest Common Factor (HCF) of two positive numbers. Euclid's Division Lemma states that:

Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

#### The Fundamental Theorem of Arithmetic

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Using this theorem, we can express every natural number as a prime number multiplication, for example, 35 = 7 × 5, 253 = 11 x 23, etc.

#### Rational and Irrational Numbers and their Decimal Expansions

Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

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### Frequently Asked Questions on NCERT Class 10 Maths Chapter 1 : Real Numbers

Q1. Explain the Graphical Method of Solutions of a Pair of Linear Equations.

Answer: Two lines are represented by the graph of a pair of linear equations with two variables.

1. A system is considered to have a unique solution if the graphs of its two equations intersect at a single point.
2. A system is considered to have no solution if the graphs of two equations are parallel lines.
• When the graphs of two equations in a system are two coincident lines, the system is said to have an unlimited number of solutions, indicating that it is consistent and dependent.

Q2. Are Real Numbers NCERT Solutions for Class 10 Maths Chapter 1 important for exams?

Answer: Yes, Chapter 1 of NCERT Class 10 Maths Solutions is significant for the exam. Your NCERT Class 10 Maths book contains critical questions. This chapter contains both short-answer and long-answer questions. This is a foundational chapter, and the topics presented will be used in subsequent chapters. This chapter will help you improve your problem-solving abilities.

Q3. What are the Key Topics in NCERT Solutions Class 10 Maths Chapter 1?

Answer: Euclid's Division Lemma and the Fundamental Theorem of Arithmetic are two major subtopics taught in NCERT Solutions Class 10 Maths Chapter 1. Students can utilize Euclid's Division Lemma to compute the HCF of two positive integers, and the Fundamental Theorem of Arithmetic to find the HCF and LCM of two positive integers. Students can practice questions relating to both areas to have a thorough comprehension of the material.

Q4. How many exercises are in NCERT Solutions for Class 10 Maths Chapter 1?

Answer: Real Numbers is Chapter 1 of the Class 10 NCERT Maths textbook. This chapter contains a total of four exercises. Every workout has a unique PDF. The NCERT Solutions Class 10 Maths Chapter 1 Real numbers have a total of 18 well-researched problems. The 18 questions are divided into three categories: long replies, middle-level answers, and easy answers. These solutions will enable you to complete the NCERT syllabus and be prepared for the Class 10 Maths exam.

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