Euclid’s Division Lemma
1.0Master Divisibility and the HCF Algorithm in Minutes
Euclid's Division Lemma is one of the fundamental concepts in Class 10 Maths Chapter 1 – Real Numbers. It provides a mathematical method for dividing one positive integer by another and is widely used to find the Highest Common Factor (HCF) of two numbers using the Euclidean Algorithm.
The concept was introduced by the ancient Greek mathematician Euclid over 2,000 years ago and remains an important part of modern mathematics.
3.0Introduction
Euclid’s Division Lemma is one of the most important topics in Class 10 Mathematics, especially in the chapter on Real Numbers. It helps students understand how one number can be divided by another and how the remainder is always smaller than the divisor. This concept is not only useful in school exams but also forms the foundation of many number theory ideas.
In simple words, Euclid’s Division Lemma tells us that any positive integer can be written in the form of a quotient multiplied by a divisor plus a remainder. This small idea is extremely powerful because it leads to the Euclidean algorithm, which is used to find the highest common factor of two numbers.
4.0Euclid’s Division Lemma
Euclid’s Division Lemma (lemma is similar to a theorem) says that, for given two positive integers, 'a' and 'b', there exist unique integers, 'q' and 'r', such that: a = bq+r, where 0 ≤r <b.
a=bq+r
where:
- a is the dividend.
- b is the divisor.
- q is the quotient.
- r is the remainder.
- 0≤r<b
This means the remainder is always zero or less than the divisor.
The integer 'q' is the quotient and the integer 'r' is the remainder. The quotient and the remainder are unique. In simple words, Euclid's division lemma statement is that if we divide an integer by another non-zero integer, we will get a unique integer as quotient and a unique integer as remainder.
We can write the above scenario mathematically as: 'Dividend = (Divisor × Quotient) + Remainder'. The above scenario shows the way of representing the division of positive integers with the help of Euclid's division lemma.
5.0Euclid’s Division Lemma v/s Euclid's Division Algorithm
6.0How to Find HCF By Euclid's Division Lemma?
Euclid's division lemma has many uses or applications. Let's learn about two of the most important uses of this algorithm. We use Euclid's division lemma to find the HCF of large numbers which is typically difficult to calculate using basic HCF calculation techniques. The HCF of two numbers can be calculated with the help of Euclid's division lemma by following these steps.
Let's take two numbers 'c' and 'd' for which we need to find the HCF such that c > d.
- Step 1:- Apply Euclid’s division lemma to 'c' and 'd'. We can find whole numbers, 'q' and 'r' such that c = dq + r, 0 ≤ r < d.
- Step 2:- If r = 0, 'd' is the HCF of 'c' and 'd'. If r ≠ 0, apply the division lemma again to 'd' and 'r'.
- Step 3:- Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.
Since the remainder is now zero, we can stop the process. The HCF of 42 and 64 is 2. When we apply Euclid's division lemma repeatedly to find the HCF, this process is known as Euclid's division algorithm.
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8.0Supporting Study Materials
This study material, including CBSE Notes and NCERT Solutions for Chapter 1 of Class 10 Mathematics, is structured according to the latest NCERT guidelines. Complete with step-by-step numerical division tracks, remainder boundary highlights, and high-yield subjective proofs, this guide ensures absolute structural confidence for your school assessments and board examinations.
9.030-Second Quick Revision: Euclid’s Division Lemma
Here are a few important points to remember about Euclid's division lemma.
- Euclid's Division Lemma is applicable only to positive integers.
- The remainder is always less than the divisor.
- Every division can be written in the form: a=bq+r
- The quotient and remainder are unique.
- The lemma forms the basis of the Euclidean Algorithm used to calculate the HCF.
10.0Previous Year Questions (PYQs) on Euclid’s Division Lemma
Question: Find the HCF of 455 and 42.
Solution: Step 1: 455=42×10+35
Step 2: 42=35×1+7
Step 3: 35=7×5+0
Therefore, HCF = 7
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