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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.

Q5  Use Euclid’s division lemma to show that the cube of any positive integer is of the form
       9m, 9m + 1 or 9m + 8.

Answers (1)
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S Sayak

Let x be any positive integer.

It can be written in the form 3q + r where q\geq 0 and r = 0, 1 or 2

Case 1:

For r = 0 we have

x3 = (3q)3

x3 = 27q3

x3 = 9(3q3)

x3 = 9m

Case 2:

For r = 1 we have

x3 = (3q+1)3

x3 = 27q3 + 27q2 + 9q + 1

x3 = 9(3q3 + 3q2 +q) + 1

x3 = 3m + 1

Case 3:

For r = 2 we have

x3 = (3q+2)3

x3 = 27q3 + 54q2 + 36q + 8

x3 = 9(3q3 + 6q2 +4q) + 8

x3 = 3m + 8

Hence proved.

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