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Question

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 some integer m.

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Solution

Let x be any positive integer. Then, it is of the form 3q or,3q+1 or, 3q+2.

Case-I When x=3q

x3=(3q)3=27q3=9(3q3)=9m, where m=3q3

Case-II when x=3q+1

x3=(3q+1)3

x3=27q3+27q2+9q+1

x3=9q(3q2+3q+1)+1

x3=9m+1, where m=q(3q2+3q+1)

Case-III when x=3q+2

x3=(3q+2)3

x3=27q3+54q2+36q+8

x3=9q(3q2+6q+4)+8

x3=9m+8,where m=q(3q2+6q+4)

Hence, x3 is either of form 9m or 9m+1 or 9m+8.

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