Question
Thu June 28, 2012 By: Medha
 

show that the cube of any positive integer is of the form 9m, 9m+1 or 9m-1 for some integer m.

Expert Reply
Thu June 28, 2012

Let a be any positive integer and b = 3

a = 3q + r, where q and r are greater than or equal 0 and r < 3

 

Therefore, every number can be represented as these three forms. There are three cases.

 

Case 1:    When a = 3q,

a3 = 9m

 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 + 1) - 1

a3 = 9m - 1

Where m is an integer such that m = (3q3 + 6q2 + 4q + 1)

 

Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m - 1.
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