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

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