Question

.................................................

Answer 1

Let x be any positive integer, then it is of  the form 3m, 3m + 1 or 3m +2. Now, we have prove that the cube of each of these can be rewritten in the form 9q + 1 or 9q + 8.
            Now, (3m)3=27m3=9(3m3)
            = 9q, where q=3m3
            (3m+1)3=(3m)3+3(3m)2.1+3(3m).12+1
            =27m3+27m2+9m+1
            =9(3m3+3m2+m)+1
            = 9q + 1, where q=3m3+3m2+m
            and (3m+2)3=(3m)3+3(3m)2.2+3(3m).22+8
            =27m3+54m2+36m+8
            =9(3m3+6m2+4m)+8
            = 9q + 8, where q=3m3+6m2+4m