##### NCERT Solutions for Class 10 Maths | Chapter: Real Numbers | Concept: Applicatoin of Euclid's Division Algorithm

#### Question Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m +1 for some integer m.

[**Hint:** Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

#### Explanatory Answer

Let ‘x’ be any positive integer.

Let us apply Euclid’s division algorithm on ‘x’ with 3 as the divisor.

x = 3q + r , 0 ≤ r < 3

∴ x = 3q + 0 or 3q + 1 or 3q + 2

**If x = 3q**, x^{2} = 9q^{2} = 3(3q^{2})

Let 3q^{2} = m

So, x^{2} = 3m -------------(1)

**If x = 3q + 1**, x^{2} = (3q + 1)^{2} = 9q^{2} + 6q + 1

x^{2} = 3(3q^{2} + 2q) + 1

Let 3q^{2} + 2q be ‘m’

So, x^{2} = 3m + 1 --------(2)

**If x = 3q + 2**, x^{2} = (3q + 2)^{2} = 9q^{2} + 12q + 4

x^{2} = 9q^{2} + 12q + 3 + 1

= 3(3q^{2} + 4q + 1) + 1

Let 3q^{2} + 4q + 1 be m

x^{2} = 3m + 1 --------------- (3)

From (1), (2), and (3) we can conclude that the square of a positive number, x^{2} is of the form 3m or 3m + 1