Chapter 1 of the Class 10 NCERT Mathematics textbook is titled "Real Numbers." This chapter delves into fundamental concepts such as Euclid's Division Lemma, the Fundamental Theorem of Arithmetic, and the properties of rational and irrational numbers. The chapter is structured into four exercises, each focusing on specific topics. Below is a detailed overview of each exercise, including the questions and step-by-step solutions.
Exercise 1.1
This exercise focuses on applying Euclid's Division Lemma to find the Highest Common Factor (HCF) of given pairs of numbers.
1. Question 1: Use Euclid’s division algorithm to find the HCF of:
(i) 135 and 225
(ii) 196 and 38220
(iii) 867 and 255
Solution:
(i): Apply Euclid's Division Lemma: 225 = 135 × 1 + 90. Then, 135 = 90 × 1 + 45. Next, 90 = 45 × 2 + 0. Since the remainder is 0, the HCF is 45.
(ii): Apply Euclid's Division Lemma: 38220 = 196 × 195 + 0. Since the remainder is 0, the HCF is 196.
(iii): Apply Euclid's Division Lemma: 867 = 255 × 3 + 102. Then, 255 = 102 × 2 + 51. Next, 102 = 51 × 2 + 0. Since the remainder is 0, the HCF is 51.
2. Question 2: Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5, where q is some integer.
Solution:
Any integer n can be expressed as n = 6q + r, where r = 0, 1, 2, 3, 4, or 5.
Substituting r = 0, 2, or 4 gives even numbers.
Substituting r = 1, 3, or 5 gives odd numbers.
Therefore, any positive odd integer can be expressed as 6q + 1, 6q + 3, or 6q + 5.
3. Question 3: An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
Solution:
Find the HCF of 616 and 32 using Euclid's Division Algorithm.
616 = 32 × 19 + 8
32 = 8 × 4 + 0
The HCF is 8.
Therefore, the maximum number of columns is 8.
4. Question 4: Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
Solution:
Any integer n can be expressed as n = 3q, 3q + 1, or 3q + 2.
Squaring these forms gives:
(3q)² = 9q² = 3(3q²)
(3q + 1)² = 9q² + 6q + 1 = 3(3q² + 2q) + 1
(3q + 2)² = 9q² + 12q + 4 = 3(3q² + 4q + 1) + 1
Thus, the square of any positive integer is either of the form 3m or 3m + 1.
Exercise 1.2
This exercise involves problems based on the Fundamental Theorem of Arithmetic, focusing on prime factorization and finding the HCF and LCM of numbers.
1. Question 1: Express each number as a product of its prime factors:
(i) 140
(ii) 156
(iii) 3825
(iv) 5005
(v) 7429
Solution:
(i): 140 = 2 × 2 × 5 × 7
(ii): 156 = 2 × 2 × 3 × 13
(iii): 3825 = 3 × 5 × 5 × 17
(iv): 5005 = 5 × 7 × 11 × 13
(v): 7429 = 17 × 19 × 23
2. Question 2: Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers:
(i) 26 and 91
(ii) 510 and 92
(ii
i) 336 and 54
Solution:
(i):
Prime factors: 26 = 2 × 13; 91 = 7 × 13
HCF = 13; LCM = 2 × 7 × 13 = 182