Answer Calculated in the video has some mistake please check the image for the correct answer.

Last updated at June 22, 2017 by Teachoo

Answer Calculated in the video has some mistake please check the image for the correct answer.

Transcript

Ex 1.1 , 5 Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+ 1 or 9m+ 8. As per Euclid’s Division Lemma If a and b are 2 positive integers, then a = bq + r where 0 ≤ r < b Let positive integer be a And b = 3 Hence a = 3q + r where ( 0 ≤ r < 3) r is an integer greater than or equal to 0 and less than 3 hence r can be either 0 , 1 or 2 If r = 0 Our equation becomes a = 3q + r a = 3q + 0 a = 3q Cubing both sides a3 = (3q)3 a3 = 27q3 a3 = 9(3q3) a3 = 9m Where m = 3q3 If r = 1 Our equation becomes a = 3q + r a = 3q + 1 Cubing both sides a3 = (3q + 1)3 a3 = (3q)3 + 13 + 3 × 3q × 1(3q + 1) a3 = 27q3 + 1 + 9q × (3q + 1) a3 = 27q3 + 1 + 27q2 + 9q a3 = 27q3 + 27q2 + 9q + 1 a3 = 9(3q3 + 3q2 + q) + 1 a3 = 9m + 1 Where m = 3q3 + 3q2 + q If r = 2 Our equation becomes a = 3q + r a = 3q + 2 Cubing both sides a3 = (3q + 2)3 a3 = (3q)3 + 23 + 3 × 3q × 2(3q + 2) a3 = 27q3 + 8 + 18q × (3q + 2) a3 = 27q3 + 8 + 18q2 + 6q a3 = 27q3 + 54q2 + 36q + 8 a3 = 9(3q3 + 6q2 + 4q) + 8 a3 = 9m + 8 Where m = 3q3 + 6q2 + 4q Hence, cube of any positive number can be expressed of the form 9m or 9m + 1 or 9m + 8 Hence proved

Class 10

Important Questions for Exam - Class 10

- Chapter 1 Class 10 Real Numbers
- Chapter 2 Class 10 Polynomials
- Chapter 3 Class 10 Pair of Linear Equations in Two Variables
- Chapter 4 Class 10 Quadratic Equations
- Chapter 5 Class 10 Arithmetic Progressions
- Chapter 6 Class 10 Triangles
- Chapter 7 Class 10 Coordinate Geometry
- Chapter 8 Class 10 Introduction to Trignometry
- Chapter 9 Class 10 Some Applications of Trignometry
- Chapter 10 Class 10 Circles
- Chapter 11 Class 10 Constructions
- Chapter 12 Class 10 Areas related to Circles
- Chapter 13 Class 10 Surface Areas and Volumes
- Chapter 14 Class 10 Statistics
- Chapter 15 Class 10 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.