Using Division Lemma.

 

using euclid's division lemma
a= bq+r where r is greater or equal to zero and less than b
b=3
so the possible values of r will be 0, 1, 2

so possible values of r are 0, 1and 2.

Now consider consider 3n,3n+1, 3n+2.

cubes are 27n^3, 27n^3+27n^2+9n+1, 27n^3+54n^2+36n+8
i.e 27n^3, 9(3n^3+3n^2+n)+1, 9(3n^3+6n^2+4n)+8
which all have the form 9m,9m+1 or 9m+8