Answer with explanation:
It is given that , n is Odd integer.
If , n is odd, then it can be Written as with the help of Euclid division lemma
→ n= 2 p +1, as 2 p is even , and adding 1 to it converts it into Odd.
As Euclid lemma states that for any three integers, a ,b and c ,when a is divided by b, gives quotient c and remainder r , then it can be written as:
a= b c+r, →→0≤r<b
Now, n² will be of the form
(2 p +1)²=(2 p)²+2 × 2 p×1+ (1)²
=4 p²+4 p +1
⇒Multiplying any positive or negative Integer by 4, gives Even integer and sum or Difference of two even integer is always even.
So, 4 p²+4 p, will be an even term.But Adding , 1 to it converts it into Odd Integer.
Hence, if n is an Odd number then , n² will be also odd.