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# Prove that the Square of Any Positive Integer is of the Form 4q Or 4q + 1 for Some Integer Q. - CBSE Class 10 - Mathematics

#### Question

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

#### Solution

By Euclid’s division Algorithm

a = bm + r, where 0 ≤ r ≤ b

Put b = 4

a = 4m + r, where 0 ≤ r ≤ 4

If r = 0, then a = 4m

If r = 1, then a = 4m + 1

If r = 2, then a = 4m + 2

If r = 3, then a = 4m + 3

Now, (4m)2 = 16m2

= 4 × 4m2

= 4q where q is some integer

(4m + 1)2 = (4m)2 + 2(4m)(1) + (1)2

= 16m2 + 8m + 1

= 4(4m2 + 2m) + 1

= 4q + 1 where q is some integer

(4m + 2)2 = (4m)2 + 2(4m)(2)+(2)2

= 16m2 + 24m + 9

= 16m2 + 24m + 8 + 1

= 4(4m2 + 6m + 2) + 1

= 4q + 1, where q is some integer

Hence, the square of any positive integer is of the form 4q or 4q + 1 for some integer m

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: 8 | Page no. 10

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Solution Prove that the Square of Any Positive Integer is of the Form 4q Or 4q + 1 for Some Integer Q. Concept: Euclid’s Division Lemma.
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