Use Euclid\'s division lemma to show that the cube of any positive integer is the form 9m,9m+1or9m+8

Use Euclid\'s division lemma to show that the cube of any positive int

1.	Use Euclid\'s division lemma to show that the cube of any positive integer is the form 9m,9m+1or9m+8
Answer» Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.So, we have the following cases :Case I : When x = 3q.then, x3 = (3q)3 = 27q3 = 9 (3q3) = 9m, where m = 3q3.Case II : When x = 3q + 1then, x3 = (3q + 1)3= 27q3 + 27q2 + 9q + 1= 9 q (3q2 + 3q + 1) + 1= 9m + 1, where m = q (3q2 + 3q + 1)Case III. When x = 3q + 2then, x3 = (3q + 2)3= 27 q3 + 54q2 + 36q + 8= 9q (3q2 + 6q + 4) + 8= 9 m + 8, where m = q (3q2 + 6q + 4)Hence, x3 is either of the form 9 m or 9 m + 1 or, 9 m + 8.