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use Euclid's division lemma to show that the cube of any positive integer is of the form 9m , 9m+1 , or 9m+8 for any integer m .
Let a be any positive integer and b = 3
a = 3q + r, where q ? 0 and 0 ? r < 3
So a = 3q or 3q + 1 or 3q + 2

Therefore, every number can be represented as these three forms. There are three cases.
Case 1: When a = 3q, 
a3 = (3q)3 = 27q3 = 9(3q3) = 9
Where m is an integer such that m = 3q3 
Case 2: 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) 
Case 3: When a = 3q + 2,
a3 = (3q +2)3
a3 = 27q3 + 54q2 + 36q + 8 
a3 = 9(3q3 + 6q2 + 4q) + 8
a3 = 9m + 8
Where m is an integer such that m = (3q3 + 6q2 + 4q)
Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.


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