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

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 + 1

then, 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 + 2

then, 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.