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Prove that the Square of Any Positive Integer is of the Form 3m Or, 3m + 1 but Not of the Form 3m +2. - CBSE Class 10 - Mathematics

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Question

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2.

Solution

By Euclid’s division algorithm

a = bq + r, where 0 ≤ r ≤ b

Put b = 3

a = 3q + r, where 0 ≤ r ≤ 3

If r = 0, then a = 3q

If r = 1, then a = 3q + 1

If r = 2, then a = 3q + 2

Now, (3q)2 = 9q2

= 3 × 3q2

= 3m, where m is some integer

(3q + 1)2 = (3q)2 + 2(3q)(1) + (1)2

= 9q2 + 6q + 1

= 3(3q2 + 2q) + 1

= 3m + 1, where m is some integer

(3q + 2)2 = (3q)2 + 2(3q)(2) + (2)2

= 9q2 + 12q + 4

= 9q2 + 12q + 4

= 3(3q2 + 4q + 1) + 1

= 3m + 1, hwrer m is some integer

Hence the square of any positive integer is of the form 3m, or 3m +1

But not of the form 3m + 2

  Is there an error in this question or solution?

APPEARS IN

 RD Sharma Solution for 10 Mathematics (2018 to Current)
Chapter 1: Real Numbers
Ex. 1.10 | Q: 7 | Page no. 10

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Solution Prove that the Square of Any Positive Integer is of the Form 3m Or, 3m + 1 but Not of the Form 3m +2. Concept: Euclid’s Division Lemma.
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