Key Concepts and Applications of Euclid’s Division Lemma for Exam Success
FAQs on Master Euclid’s Division Lemma with Step-by-Step Guidance for CBSE 2025-26
1. Is Euclid's Division Lemma an important topic for the CBSE Class 10 Maths board exam 2025-26?
No, as per the rationalised syllabus for the CBSE Class 10 Maths 2025-26 session, the topic of Euclid's Division Lemma has been removed from Chapter 1, Real Numbers. While it was previously a source of important questions, the current curriculum focuses on the Fundamental Theorem of Arithmetic for finding the HCF and LCM of numbers. Students should focus on practice questions related to the Fundamental Theorem of Arithmetic for their board exam preparation.
2. If it's no longer in the syllabus, why do some resources still list important questions from Euclid's Division Lemma?
While not required for the CBSE 2025-26 board exams, understanding Euclid's Division Lemma is beneficial for building a strong conceptual foundation in number theory. The logical process it teaches is valuable for various competitive exams like Maths Olympiads. Learning how it's used to find the HCF helps in appreciating the structure of the number system, even if direct questions are not asked in board exams.
3. What were the typical types of important questions asked from Euclid's Division Lemma in previous years' board exams?
In past examinations, questions from this topic were frequently asked and generally fell into two main categories:
Application-based questions (2-3 marks): These required using Euclid's division algorithm to find the HCF of two or three positive integers, including word problems that model a situation requiring the HCF.
Proof-based questions (3-4 marks): These were considered Higher Order Thinking Skills (HOTS) questions. A typical example would be to 'Show that any positive odd integer is of the form 4q + 1 or 4q + 3', which required a deep understanding of the lemma.
4. What is the most crucial application of Euclid's Division Lemma when it comes to solving problems?
The most important and direct application of Euclid's Division Lemma is in the Euclid's Division Algorithm. This algorithm provides a highly efficient and systematic method to compute the Highest Common Factor (HCF) of two positive integers. For very large numbers, this algorithm is often much faster and less error-prone than the prime factorisation method.
5. What is the fundamental difference between Euclid's Division Lemma and Euclid's Division Algorithm?
It's a common point of confusion, but the distinction is important for conceptual clarity.
The Lemma is the proven mathematical statement itself: For any two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r, where 0 ≤ r < b.
The Algorithm is the practical procedure or sequence of steps that uses the lemma repeatedly to find the HCF of 'a' and 'b'. The algorithm is the application; the lemma is the foundation it stands on.
6. How can one prove that the square of any positive integer is of the form 3m or 3m+1 using the lemma?
This is a classic HOTS question type. The strategy involves using the lemma to represent any positive integer 'a' in terms of a divisor, in this case, 3. The steps are:
Apply Euclid's Division Lemma to 'a' with the divisor b=3. 'a' can be written as 3q, 3q+1, or 3q+2.
Square each of these three cases individually: (3q)², (3q+1)², and (3q+2)².
Rearrange the algebraic expressions. For example, (3q)² = 9q² = 3(3q²), which is of the form 3m. Similarly, (3q+1)² = 9q²+6q+1 = 3(3q²+2q)+1, which is of the form 3m+1. Repeating this for all cases proves the statement.
7. How does the repeated application of Euclid's Division Lemma in the algorithm guarantee that we will find the HCF?
The guarantee comes from a core property: the HCF of two numbers (a, b) is the same as the HCF of the smaller number and the remainder (b, r). Since the remainder 'r' is always smaller than the divisor 'b', each step of the algorithm replaces a larger problem with a smaller one. This process must end because the remainders are decreasing whole numbers and will eventually become zero. The last non-zero remainder is the HCF because at that stage, it divides the previous divisor perfectly, and by the chain of logic, it must also be the greatest common divisor of the original two numbers.
