Use Euclid's Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m - Mathematics

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Use Euclid's Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m

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Solution

Let a and b be two positive integers such that a is greater than b; then:

a = bq + r; where q and r are positive integers and 0 ≤ r < b.

Taking b = 3, we get:

a = 3q + r; where 0 ≤ r < 3

⇒ Different values of integer a are 3q, 3q + 1 or 3q + 2.

Cube of 3q = (3q)3 = 27q3 = 9(3q3) = 9m; where m is some integer.

Cube of 3q + 1 = (3q + 1)3

= (3q)3 + 3(3q)2 x 1 + 3(3q) x 12 + 13

[Q (q + b)3 = a3 + 3a2b + 3ab2 + 1]

= 27q3 + 27q2 + 9q + 1

= 9(3q3 + 3q2 + q) + 1

= 9m + 1; where m is some integer.

Cube of 3q + 2 = (3q + 2)3

= (3q)3 + 3(3q)2 x 2 + 3 x 3q x 22 + 23

= 27q3 + 54q2 + 36q + 8

= 9(3q3 + 6q2 + 4q) + 8

= 9m + 8; where m is some integer.

Cube of any positive integer is of the form 9m or 9m + 1 or 9m + 8.

Hence the required result.

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Chapter 1: Real Numbers - Exercise 1.1 [Page 7]

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NCERT Mathematics Class 10
Chapter 1 Real Numbers
Exercise 1.1 | Q 5 | Page 7

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