Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. (CBSE 2009)

According to Euclid's division lemma, a = bq + r

By taking,’ a’ as any positive integer and b = 3.

Applying Euclid’s algorithm

a = 3q+r

Here, r =  remainder
q = quotient 

Now, as 3 is the divisor, the possible values of remainder are 0,1 and 2
as, 0 ≤ r < a

Possible values of r are 0, 1 and 2.

So, total possible forms are

when a = 3 q,

a³ = (3 q)³ = 27 q³ = 9(3 q³) = 9 m,

where m is an integer such that m = 3 q³
[as, q is an integer m = 3 q³ is also an integer ] 

when a = 3 q + 1,

a³ = (3 q+1)³

a³ = 27 q³ + 27 q² + 9q + 1

a³ = 9 (3 q³ + 3 q² + q) + 1

a³ = 9 m+1

where m is an integer such that m = (3 q³ + 3 q² + q)

when a = 3 q + 2,

a³ = (3 q + 2)³

a³ = 27 q³ + 54 q² + 36 q + 8

a³ = 9 (3 q³ + 6 q² + 4 q) + 8

a³ = 9 m + 8

Hence,

The cube of any positive integer is of the form 9 m , 9 m+1, 9 m+8