**4.** 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 . Now square each of these and show that they can be rewritten in the form 3m or 3m+1]

**Solution:**

Let x be any positive integer and b=3.

According to **Euclid's division lemma**, we can say that

Therefore, all possible values of **x** are:

or

Now lets square each one of them one by one.

(i)

Let be some integer, we get

(ii)

Let be some integer, we get

(iii)

Let be some integer, we get

Hence, square of any positive integer is either of the form 3m or 3m+1 for some integer m.