NCERT Solutions for Class 10 Chapter 1 Real numbers Exercise 1.1 Q4

Extra Questions For Real Numbers
NCERT Solutions for Class 10 Chapter 1 Real numbers Exercise 1.1 Q4
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 x be any positive integer and b=3.
According to Euclid’s division lemma, we can say that $x=3q+r,0 \le r

Therefore, all possible values of x are:
x=3q,(3q+1) or (3q+2)

Now lets square each one of them one by one.

Case 1:
[latex](3q)^2=9q^2$

Let $m=3q^2$ be some integer, we get $9q^2=3 \times 3q^2 = 3m$

Case 2: $(3q+1)^2=9q^2+6q+1=3(3q^2+2q)+1$
Let $m=3q^2+2q$ be some integer, we get $3q+1)^2=3m+1$ $3q+2)^2=9q^2+4+12q=9q^2+12q+3+1=3(3q^2+4q+1)+1$

Case 3:
Let $m=(3q^2+4q+1)$ be some integer, we get $(3q+2)^2=3m+1$

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