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

By taking,’ a’ as  any positive integer and b = 3.
Applying Euclid’s  algorithm
a = 3q+r                
Here,
So, total possible forms are

Therefore, every number can be represented as these three forms.
when a = 3q,
a³ = (3q)³ = 27q³ = 9(3q³) = 9m,
where m is an integer such that m = 3q³
when a = 3q + 1,
a³ = (3q+1)³
a³ = 27q³+27q²+9q+1
a³ = 9(3q³+3q²+q)+1
a³ = 9m+1
where m is an integer such that m = (3q³+3q²+q)
when a = 3q+2,
a³ = (3q+2)³
a³ = 27q³+54q²+36q+8
a³ = 9(3q³+6q²+4q)+8
a3 = 9m+8
Hence,
The cube of any positive integer is of the form 9m , 9m+1, 9m+8