How to Use Euclid’s Division Lemma to Find HCF (With Solved Examples)
The concept of Euclid Division Lemma plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Euclid Division Lemma?
Euclid Division Lemma is a statement in mathematics that says for any two positive integers, say a and b, there exist unique whole numbers q (quotient) and r (remainder) such that a = bq + r, where r is greater than or equal to 0 and less than b (0 ≤ r < b). You’ll find this concept applied in areas such as HCF calculation, prime factorization, and number system questions.
Key Formula for Euclid Division Lemma
Here’s the standard formula: \( a = bq + r, \ 0 \leq r < b \)
a = Dividend, b = Divisor, q = Quotient, r = Remainder
Cross-Disciplinary Usage
Euclid Division Lemma is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for board exams, JEE, or NEET will see its relevance in number theory, cryptography, and divisibility problems.
Step-by-Step Illustration
Let's see how to use the lemma to find the HCF of two numbers, 210 and 55:
- Start with the largest number: 210 and 55.
210 ÷ 55 gives quotient 3 and remainder 45.
So, 210 = 55 × 3 + 45 - Now use 55 and 45.
55 = 45 × 1 + 10
- Next, use 45 and 10.
45 = 10 × 4 + 5
- Now use 10 and 5.
10 = 5 × 2 + 0
The remainder is now 0, so the last divisor, 5, is the HCF of 210 and 55.
Euclid Division Lemma vs Euclid Division Algorithm
| Feature | Euclid Division Lemma | Euclid Division Algorithm |
|---|---|---|
| Type | Mathematical statement (lemma/proved fact) | Series of steps/method to apply lemma |
| Purpose | Proves that dividend can be expressed as product plus remainder | Uses lemma repeatedly to find HCF/GCD |
| Exam Focus | Statement and proof questions | Solve for HCF, GCD, divisibility |
Try These Yourself
- Use Euclid Division Lemma to find the HCF of 81 and 675.
- Write the lemma form for 100 divided by 17.
- Solve: If a = 57, b = 8, express in form a = bq + r and find q and r.
Frequent Errors and Misunderstandings
- Confusing remainder r as equal to b instead of strictly less than b (r < b).
- Forgetting to stop when remainder becomes zero during HCF steps.
- Mixing up the terms "lemma" (statement) and "algorithm" (procedure).
Relation to Other Concepts
The idea of Euclid Division Lemma connects closely with topics such as Prime Numbers, Factors and Multiples, and the Fundamental Theorem of Arithmetic. Mastering this helps you understand divisibility and factorization in later chapters.
Classroom Tip
A simple way to remember the division lemma is: "Dividend = (Divisor × Quotient) + Remainder". Vedantu’s teachers often teach students to keep checking that the remainder is always smaller than the divisor for each division step.
We explored Euclid Division Lemma—from its definition, formula, examples, common mistakes, and how it links to other Maths topics. Mastering this will help you with HCF, divisibility, and more advanced problems. Continue practicing with Vedantu's free lessons to gain confidence and accuracy in maths!
Related Reading: Highest Common Factor (HCF), Prime Numbers, Factors and Multiples, Fundamental Theorem of Arithmetic
FAQs on Euclid Division Lemma: Concept, Formula, and Applications
1. What is Euclid's Division Lemma?
Euclid's Division Lemma is a fundamental theorem in number theory. It states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This means any integer a can be expressed as a multiple of another integer b plus a remainder r, which is always smaller than b.
2. How do I use Euclid's Division Lemma to find the Highest Common Factor (HCF)?
Euclid's Division Lemma forms the basis of the Euclidean algorithm for finding the HCF. The steps are:
- Apply the lemma to the two numbers, identifying the quotient (q) and remainder (r).
- If the remainder (r) is 0, the HCF is the divisor (b) from the previous step.
- If the remainder is not 0, replace the larger number (a) with the divisor (b) and the smaller number (b) with the remainder (r).
- Repeat steps 1-3 until the remainder is 0. The last non-zero remainder is the HCF.
3. What is the difference between Euclid's Lemma and Euclid's Algorithm?
Euclid's Division Lemma is a mathematical statement—a proven fact—that establishes the relationship between two integers through division. Euclid's Algorithm is a procedure, a step-by-step method that uses the lemma repeatedly to find the Highest Common Factor (HCF) of two integers.
4. What are some applications of Euclid's Division Lemma beyond finding the HCF?
Euclid's Division Lemma has broader applications in number theory, including proving divisibility rules, solving problems related to remainders, and establishing properties of prime numbers. It also underpins concepts in cryptography and modular arithmetic.
5. Can you provide a solved example using Euclid's Lemma to find the HCF?
Let's find the HCF of 48 and 18 using Euclid's algorithm:
- 48 = 18 × 2 + 12 (Remainder is 12)
- 18 = 12 × 1 + 6 (Remainder is 6)
- 12 = 6 × 2 + 0 (Remainder is 0)
Since the remainder is 0, the HCF is the last non-zero remainder, which is 6.
6. What happens if 'b' divides 'a' exactly in Euclid's Division Lemma?
If b divides a exactly, the remainder r becomes 0. The equation then simplifies to a = bq, indicating that a is a multiple of b. In the context of the Euclidean algorithm, this means b is the HCF.
7. Why is Euclid's Division Lemma called a 'lemma' and not a 'theorem'?
A lemma is a smaller, proven statement that serves as a stepping stone for proving larger theorems. Euclid's Division Lemma is primarily used as a tool to prove other significant results in number theory, hence its designation as a lemma.
8. How is the uniqueness of 'q' and 'r' proven in Euclid's Division Lemma?
The uniqueness of q and r is proven through contradiction. Assume there are two different pairs of (q, r) that satisfy the equation. By subtracting the two equations and manipulating the result, a contradiction is reached, demonstrating that only one unique pair can exist.
9. How does Euclid's Division Lemma relate to prime numbers and factorization?
Euclid's Division Lemma is crucial for understanding prime factorization. The lemma helps in establishing the fundamental theorem of arithmetic, which states that every integer greater than 1 can be uniquely expressed as a product of prime numbers. The process of prime factorization relies on repeated application of the division lemma.
10. Are there any limitations to Euclid's Division Lemma or Algorithm?
The main limitation is that it applies only to positive integers. It cannot directly be used with negative integers or rational numbers. However, extensions and modifications exist to handle other number systems.
11. What is the significance of Euclid's Division Lemma in mathematics?
Euclid's Division Lemma is a cornerstone of number theory. Its importance lies in its simplicity and wide-ranging applications. It provides a foundational tool for understanding the structure and properties of integers, underpinning many advanced concepts and theorems.
