Question

use Euclids division lemma to show that the cube of any positive integer is in the form of 9M ,9M + 1or9M + 8 ​

Answer 1

Let n be any positive integer. Then , it is of the form 3q or 3q + 1 or 3q + 2.

Case 1: 

When n = 3q

 = 9 m + 2, where m = 3q3, is an integer.

Case 2:

When n = 3q + 1

 = 9 m + 3, where m = 3q3 + 3q2 + q, is an integer 

Case 3:

When n = 3q + 2

 = 9 m + 1, where m = 3q3 + 6q2 + 4q + 1, is an integer.

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Answer 2

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