Question

. Use Euclid's division lemma to show that the cube of any positive integer is of the form
9m, 9m + 1 or 9m +8.


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Answer 1

Step-by-step explanation:

here is your answer:

Let us consider a and b where a be any positive number and b is equal to 3.

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemma

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + r

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + r

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomes

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3q

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3qCubing both the sides

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3qCubing both the sidesa3 = (3q)3

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3qCubing both the sidesa3 = (3q)3a3 = 27 q3

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3qCubing both the sidesa3 = (3q)3a3 = 27 q3a3 = 9 (3q3)

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3qCubing both the sidesa3 = (3q)3a3 = 27 q3a3 = 9 (3q3)a3 = 9m

Let us consider a and b where a be any positive number and b is equal to 3.According to Euclid’s Division Lemmaa = bq + rwhere r is greater than or equal to zero and less than b (0 ≤ r < b)a = 3q + rso r is an integer greater than or equal to 0 and less than 3.Hence r can be either 0, 1 or 2.Case 1: When r = 0, the equation becomesa = 3qCubing both the sidesa3 = (3q)3a3 = 27 q3a3 = 9 (3q3)a3 = 9mwhere m = 3q3