Math Class 10 Exercise 1.1

Conquering Math Class 10 Exercise 1.1: A Comprehensive Guide

Are you a Class 10 student grappling with Exercise 1.1 in your math textbook? This comprehensive guide will walk you through each problem, providing clear explanations, step-by-step solutions, and helpful tips to build your understanding of the underlying mathematical concepts. We'll tackle the common challenges faced by students, ensuring you gain confidence and master this crucial section. This exercise typically focuses on the fundamentals of real numbers, specifically dealing with Euclid's division lemma and the fundamental theorem of arithmetic. Let's dive in!

Introduction: Understanding the Basics

Exercise 1.1 in most Class 10 math textbooks introduces the core concepts of real numbers. This foundational chapter lays the groundwork for more advanced topics later in the year. The exercises within this section usually focus on two key theorems:

• Euclid's Division Lemma: This lemma 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. Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder.

• Fundamental Theorem of Arithmetic: This theorem, also known as the unique factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers (ignoring the order of the factors).

Understanding these two theorems is crucial to successfully completing Exercise 1.1. Let's move on to the step-by-step solutions and explanations. We will assume a typical exercise set containing problems involving finding the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) using Euclid's division lemma and prime factorization.

Step-by-Step Solutions & Explanations: Typical Problem Types

Since the exact questions in Exercise 1.1 vary slightly depending on the textbook, we will cover the most common types of problems found in this section.

Problem Type 1: Applying Euclid's Division Lemma

This type of problem often asks you to use Euclid's division lemma to find the HCF of two numbers. Let's illustrate with an example:

Example: Find the HCF of 4052 and 12576 using Euclid's division lemma.

Solution:

1. Apply the lemma repeatedly: We start by dividing the larger number (12576) by the smaller number (4052). 12576 = 4052 × 3 + 420

2. Replace the larger number: Now, we replace the larger number (12576) with the smaller number (4052) and the smaller number with the remainder (420). We continue this process. 4052 = 420 × 9 + 212

3. Repeat until the remainder is 0: 420 = 212 × 1 + 208 212 = 208 × 1 + 4 208 = 4 × 52 + 0

4. The HCF is the last non-zero remainder: The last non-zero remainder is 4. Therefore, the HCF of 4052 and 12576 is 4.

Problem Type 2: Finding the HCF and LCM using Prime Factorization

This problem type requires you to find the prime factorization of the given numbers to determine their HCF and LCM.

Example: Find the HCF and LCM of 12 and 18 using prime factorization.

Solution:

1. Find the prime factorization of each number: 12 = 2² × 3 18 = 2 × 3²

2. Identify common factors: The common factors are 2 and 3.

3. HCF: The HCF is the product of the lowest powers of the common factors: HCF(12, 18) = 2 × 3 = 6

4. LCM: The LCM is the product of the highest powers of all the factors present in the numbers: LCM(12, 18) = 2² × 3² = 4 × 9 = 36

Problem Type 3: Proofs and Conceptual Questions

Some problems in Exercise 1.1 might involve proving certain properties related to Euclid's division lemma or the fundamental theorem of arithmetic. These questions test your deeper understanding of the concepts. For example:

• Prove that the square of any positive integer is of the form 3k or 3k + 1 for some integer k. This requires you to apply Euclid's division lemma with a divisor of 3 and analyze the possible remainders.

• Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. Similar to the previous example, this involves applying the division lemma with a divisor of 4.

Problem Type 4: Word Problems

These problems present real-world scenarios that require the application of HCF and LCM.

Example: A rectangular field is 72m long and 48m wide. We want to divide the field into square plots of equal size. What is the largest possible size of each square plot?

Solution: To find the largest possible size, we need to find the HCF of 72 and 48. Using prime factorization:

72 = 2³ × 3² 48 = 2⁴ × 3

HCF(72, 48) = 2³ × 3 = 24

The largest possible size of each square plot is 24m.

Explanation of Key Concepts

Euclid's Division Lemma: This lemma is fundamental to understanding the concept of divisibility. It provides a systematic way to find the HCF of two numbers. The repeated application of the lemma, as demonstrated in the examples above, leads to the Euclidean algorithm, an efficient method for finding the greatest common divisor.

Fundamental Theorem of Arithmetic: This theorem establishes the uniqueness of prime factorization. It means that regardless of how you factorize a number into primes, you will always get the same set of prime factors (although their order might be different). This property is extremely important in number theory and many other areas of mathematics. It's the basis for finding the HCF and LCM using the prime factorization method.

Frequently Asked Questions (FAQ)

• Q: What is the difference between HCF and LCM?

  • A: The HCF (Highest Common Factor), also known as the greatest common divisor (GCD), is the largest number that divides both given numbers without leaving a remainder. The LCM (Lowest Common Multiple) is the smallest number that is a multiple of both given numbers.

• Q: Can I use a calculator to find the HCF and LCM?

  • A: While calculators can compute HCF and LCM directly, understanding the methods behind these calculations (Euclid's division lemma and prime factorization) is crucial for solving more complex problems and developing a strong mathematical foundation.

• Q: What if I get a negative remainder when applying Euclid's division lemma?

  • A: The remainder r in Euclid's lemma must be non-negative and less than the divisor b (0 ≤ r < b). If you obtain a negative remainder, adjust it by adding the divisor until you get a positive remainder within the specified range.

• Q: Why is the Fundamental Theorem of Arithmetic important?

  • A: The fundamental theorem of arithmetic is the cornerstone of many number-theoretic results. Its uniqueness ensures consistency in calculations involving prime factorization and is essential in various mathematical applications.

Conclusion

Mastering Exercise 1.1 in your Class 10 math textbook is a crucial step in building a solid foundation in number theory. By understanding Euclid's division lemma and the fundamental theorem of arithmetic, and practicing the different problem types explained above, you'll develop the skills and confidence to tackle more advanced mathematical concepts. Remember that consistent practice and a thorough understanding of the underlying principles are key to success. Don't hesitate to review the examples and explanations multiple times, and always seek clarification from your teacher or tutor if you encounter any difficulties. Good luck, and happy problem-solving!