CBSE Class 10 Maths Chapter 1: Real Numbers NCERT Solutions

Q: What is Euclid's division algorithm?

Euclid's division algorithm is a method used to find the Highest Common Factor (HCF) of two positive integers. It is based on the principle that the HCF of two numbers does not change if the larger number is replaced by its difference with the smaller number, or more efficiently, by the remainder when the larger number is divided by the smaller number.

Q: How do I find the HCF using Euclid's division algorithm for 135 and 225?

You apply the division lemma repeatedly. First, divide 225 by 135 to get 225 = 135 × 1 + 90. Then, divide 135 by the remainder 90 to get 135 = 90 × 1 + 45. Finally, divide 90 by the remainder 45 to get 90 = 45 × 2 + 0. The last non-zero remainder, 45, is the HCF.

Q: Why must an odd integer be of the form 6q+1, 6q+3, or 6q+5?

When any integer is divided by 6, the possible remainders are 0, 1, 2, 3, 4, or 5. So, any integer can be written as 6q, 6q+1, 6q+2, 6q+3, 6q+4, or 6q+5. An odd integer is not divisible by 2. Among these forms, 6q, 6q+2, and 6q+4 are always even. Therefore, only 6q+1, 6q+3, and 6q+5 can represent odd integers.

Q: What is the significance of the remainder in Euclid's division algorithm?

The remainder is crucial because it becomes the new divisor in the next step of the algorithm. The process continues until the remainder becomes zero. The divisor at that point is the HCF of the original two numbers.

Q: Can Euclid's division algorithm be used for negative integers?

The standard statement of Euclid's division algorithm is for positive integers. However, the concept can be extended, but typically for Class 10, it is applied to positive integers to find their HCF.

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 10 PDF

This comprehensive guide provides NCERT Solutions for Class 10 Mathematics, Chapter 1: Real Numbers. It focuses on applying Euclid's division algorithm to find the Highest Common Factor (HCF) of various number pairs, demonstrating the step-by-step process. The solutions also delve into proving that any positive odd integer can be expressed in the specific forms of 6q+1, 6q+3, or 6q+5, where q is an integer. This detailed explanation helps students understand the underlying principles of number theory and divisibility. These solutions are designed to clarify complex concepts, provide a structured approach to problem-solving, and serve as an excellent resource for exam revision, ensuring students are well-prepared for their assessments.

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Board	CBSE
Class	Class 10
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 1

Chapter summary

Chapter 1 of the NCERT Class 10 Mathematics textbook introduces Real Numbers. This section provides solutions for Exercise 1.1, focusing on the application of Euclid's division algorithm to calculate the HCF of given numbers. It also includes a proof demonstrating the form of positive odd integers using the division lemma. The solutions offer a clear, step-by-step approach to mastering these fundamental concepts in number theory.

Learning outcomes

Understand and apply Euclid's division algorithm to find the HCF of numbers.
Solve problems involving the HCF of pairs of integers.
Prove that any positive odd integer can be represented in the form 6q+1, 6q+3, or 6q+5.
Apply the division lemma to analyze the properties of integers.

Topics covered

Paper topics

Real Numbers
Euclid's Division Algorithm
Highest Common Factor (HCF)
Properties of Integers
Odd and Even Integers
Divisibility Rules

Important topics

Euclid's Division Algorithm
Finding HCF using Euclid's Algorithm
Representing Odd Integers
Application of Division Lemma

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Questions and Solutions

Question 1

Use Euclid's division algorithm to find the HCF of:

135 and 225
196 and 38220
867 and 255

Solution:

We will use Euclid's division algorithm to find the HCF for each pair of numbers.

For 135 and 225:
Since 225 is greater than 135, we apply the division lemma:

$225 = 135 \times 1 + 90$

The remainder is 90, which is not 0. So, we apply the division lemma to 135 and 90:

$135 = 90 \times 1 + 45$

The remainder is 45, which is not 0. So, we apply the division lemma to 90 and 45:

$90 = 45 \times 2 + 0$

The remainder is now 0. The divisor at this stage is 45.

Therefore, the HCF of 135 and 225 is 45.
For 196 and 38220:
Since 38220 is greater than 196, we apply the division lemma:

$38220 = 196 \times 195 + 0$

The remainder is 0 in the first step. The divisor at this stage is 196.

Therefore, the HCF of 196 and 38220 is 196.
For 867 and 255:
Since 867 is greater than 255, we apply the division lemma:

$867 = 255 \times 3 + 102$

The remainder is 102, which is not 0. So, we apply the division lemma to 255 and 102:

$255 = 102 \times 2 + 51$

The remainder is 51, which is not 0. So, we apply the division lemma to 102 and 51:

$102 = 51 \times 2 + 0$

The remainder is now 0. The divisor at this stage is 51.

Therefore, the HCF of 867 and 255 is 51.

Question 2

Show that any positive odd integer is of the form $6q + 1$ , or $6q + 3$ , or $6q + 5$ , where q is some integer.

Solution:

Let 'a' be any positive odd integer. According to Euclid's division lemma, when 'a' is divided by any positive integer 'b', we can write $a = bq + r$ , where q is the quotient and r is the remainder such that $0 \le r < b$ .

In this case, we consider the divisor $b = 6$ . So, any positive integer 'a' can be expressed in one of the following forms:

$a = 6q + 0$

$a = 6q + 1$

$a = 6q + 2$

$a = 6q + 3$

$a = 6q + 4$

$a = 6q + 5$

Now, we need to determine which of these forms represent odd integers. An integer is odd if it is not divisible by 2.

$6q$ is divisible by 2 (since $6q = 2 \times 3q$ ), so it is an even integer.
$6q + 1 = 2(3q) + 1$ . This is of the form $2k + 1$ , so it is an odd integer.
$6q + 2 = 2(3q + 1)$ , so it is an even integer.
$6q + 3 = 6q + 2 + 1 = 2(3q + 1) + 1$ . This is of the form $2k + 1$ , so it is an odd integer.
$6q + 4 = 2(3q + 2)$ , so it is an even integer.
$6q + 5 = 6q + 4 + 1 = 2(3q + 2) + 1$ . This is of the form $2k + 1$ , so it is an odd integer.

Therefore, any positive odd integer must be of the form $6q + 1$ , $6q + 3$ , or $6q + 5$ , where q is some integer.

Common mistakes

Errors in applying the division lemma steps repeatedly.
Incorrectly identifying the remainder or divisor at each step.
Misinterpreting the condition for an odd integer in the proof.
Calculation errors in arithmetic operations within the algorithm.

Revision tips

Practice each part of Question 1 thoroughly to master the HCF calculation using Euclid's algorithm.
Pay close attention to the conditions (0 <= r < b) when applying the division lemma.
Understand the logic behind expressing odd integers in the given forms for Question 2.
Review the definition of odd and even numbers and their relationship with divisibility by 2.

Practice MCQs

Q1. What is the HCF of 135 and 225 using Euclid's division algorithm?

Q2. According to Euclid's division lemma, for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where:

Q3. Which of the following forms can represent any positive odd integer?

Q4. What is the HCF of 867 and 255?

Q5. If a = 6q + r, and 'a' is an odd integer, what are the possible values for 'r'?

Frequently asked questions

What is Euclid's division algorithm?