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


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