Euclid's Division Lemma Explained Visually | Class 10 Real Numbers Chapter

Euclid's Division Lemma is a method of expressing one number as a multiple of another plus a remainder. It forms the basis of the Euclidean Algorithm for finding the HCF (Highest Common Factor) of two numbers.

Table of Contents

• What is Euclid's Division Lemma?
• Lemma Formula and Explanation
• Visual Animation of the Lemma
• Example Using Euclid’s Lemma
• Uses in Class 10 Maths
• How to Use Euclid's Lemma
• FAQs

What is Euclid's Division Lemma?

Euclid's Division Lemma is a way to divide two positive integers a and b (where a > b) in the form:

a = bq + r, where 0 ≤ r < b

This lemma is the foundation of the Euclidean algorithm for finding the HCF of two numbers, which is an essential concept in Class 10 Real Numbers chapter.

Lemma Formula and Explanation

Given two numbers a and b, we divide a by b to get a quotient q and remainder r:

a = bq + r

This statement helps us break down numbers and compute HCF easily.

Visual Animation of the Lemma

Below is an interactive animation that explains how Euclid’s Division Lemma works step-by-step:

Interactive Euclid's Division Lemma Visualizer

Enter any two positive integers a and b (a ≥ b):

Enter value of a:

Enter value of b:

This animation visually breaks the number line into segments and shows how the division leaves a remainder.

Example Using Euclid’s Lemma

Example: Find the HCF of 56 and 72 using Euclid’s Division Lemma.

1. 72 = 56 × 1 + 16
2. 56 = 16 × 3 + 8
3. 16 = 8 × 2 + 0

So, HCF(56, 72) = 8

Where is Euclid’s Lemma Used in Class 10 Maths?

• To prove numbers are co-prime
• To find HCF using Euclidean algorithm
• Used in prime factorization and number theory
• Helpful in chapter-wise formula preparation

How to Use Euclid’s Division Lemma

1. Choose two positive integers a and b (a > b)
2. Divide a by b to get quotient q and remainder r
3. Write it as a = bq + r
4. If r ≠ 0, repeat the process using b and r
5. Continue until remainder becomes 0
6. The divisor at this stage is the HCF

FAQs on Euclid’s Division Lemma

Q1. What is Euclid's Division Lemma?

It 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.

Q2. Where is Euclid’s Lemma used in CBSE Class 10 Maths?

It is used to find the HCF of numbers and solve questions in Chapter 1 – Real Numbers.

Q3. Is Euclid’s Lemma part of the Class 10 board exam syllabus?

Yes, it is part of Chapter 1 – Real Numbers and is often asked in board exams with proof-based questions.

To master more Class 10 topics, check out our NCERT video solutions, Exemplar solutions, and chapter-wise weightage guide. You can also explore our review on Oswaal Question Bank and comparison with Arihant publications.