Question

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9q or 9q + 8, where q is some integer.

Answer 1

Let a is any positive integer, then it is in the form of 3b or 3b + 1 or 3b + 2. 

Hence, following three cases are possible.

From equation (1), (2) and (3) it is clear that a³ is divisible by 9. 

Then it can be written as follows a³ = 9q 

or a³ = 9q + 1, 

or a³ = 9q + 8 cube 

Hence the cube of any positive integer is of the form 9q or (9q + 1) or (9q + 8).