17. prove the following statement: let n e z. if n is odd, then n2 is odd. proof 6 pts. 2 vî±î¶ï there fore, n and n see tue classnotes

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.

1 x=22.52

2: 65/39=604+39x

65x-39x=604

26x=604

x=604/26

×=23.23