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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Euclid's Division Algorithm
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Euclid's Division Algorithm
Last updated at May 29, 2023 by Teachoo
Answer Calculated in the video has some mistake please check the image for the correct answer.
Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
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