# 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 form of 3m or 3m + 1]

**Solution:**

Suppose that there is a positive integer ‘a’.

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

If we keep the value of 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 are a = 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.

Let “a” be any positive integer and b = 3.

Then, a = 3q + r for some integer q ≥ 0 and r = 0, 1, 2 because 0 ≤ r < 3.

Therefore, a = 3q or 3q + 1 or 3q + 2 or

(a)^{2} = (3q)^{2} or (3q + 1)^{2} or (3q + 2)^{2}

a^{2} = 3(3q^{2}) or (9q^{2} + 6q + 1) or (9q^{2} + 12q + 4)

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

a^{2} is of the form m 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.

**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

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

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

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