Use Euclid’s division lemma to show that the square of any positive integer is either of 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]

Solution:

Let's consider a positive integer ‘a’.

By Euclid’s division lemma, we know that for any two positive integers a and b, there exist unique integers q and r, such that a = bq + r, 0 ≤ r < b

Let b = 3, then 0 ≤ r < 3. So, r = 0 or 1 or 2 but it can’t be 3 because r is smaller than 3.

So, the possible values of a are 3q or 3q + 1 or 3q + 2.

Now, find the square of all the possible values of a. If q is any positive integer, then its square (let’s call it “m”) will also be a positive integer.

Now, observe carefully that the square of all the positive integers is either of form 3m or 3m + 1 for some integer m.

We know that, a = 3q or 3q + 1 or 3q + 2 

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

a² = 3(3q²) or (9q² + 6q + 1) or (9q² + 12q + 4)

Case I: a² = 3(3q²)  where, m = 3q²

Case II: a² = 3(3q² + 2q) + 1 where, m = (3q² + 2q)

Case III: a² = 3(3q² + 4q +1) + 1 where, m = (3q² + 4q +1)

Thus, we see that a² is of the form 3m or 3m + 1 where, m is any positive integer.

Hence, it can be said that the square of any positive integer is either of form 3m or 3m + 1.

☛ Check: NCERT Solutions for Class 10 Maths Chapter 1

Video Solution:

Use Euclid’s division lemma to show that the square of any positive integer is either of 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]

NCERT Solutions Class 10 Maths - Chapter 1 Exercise 1.1 Question 4

Summary:

Using Euclid's division lemma, it can be proved that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

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