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

We know that according to Euclid's division lemma:
a = bq + r So, we have the following cases:

Case I When a = 3q

In this case, we have

a³ = (3q)³ = 27q³ = 9(3q³ ) = 9m, where m = 3q³

Case II When a = 3q + 1

In this case, we have

a³ = (3q + 1)³

⇒27q³ + 27q² + 9q + 1

⇒9q(3q² + 3q + 1) + 1

⇒a³ = 9m + 1, where m = q(3q² + 3q + 1)

Case III When a = 3q + 2

In this case, we have

a³ = (3q + 1)³

⇒27q³ + 54q² + 36q + 8

⇒9q(3q² + 6q + 4) + 8

⇒a³ = 9m + 8, where m = q(3q² + 6q + 4)

Hence, a³ is the form of 9m or, 9m + 1 or, 9m + 8