Use the method of direct proof to prove the following statem

Use the method of direct proof to prove the following statements

a) If x is an odd integer, then x 3 is odd

b) Suppose x, y Z. If x and y are odd, then x y is odd.

c) Every odd integer is a difference of two squares. (Example 7 = 4 2 3 2 , etc.)

x is odd integer so we can write, x=2k+1 for some integer k

x^3=(2k+1)^3=8k^3+3*2k(2k+1)+1=8k^3+12k^2+6k+1 =2(4k^3+6k^2+3k)+1

hence x^3 is odd

x,y are odd so we can write

x=2m+1,y=2n+1 for some integers m,n

xy=(2m+1)(2n+1)=4mn+2m+2n+1=2(2mn+m+n)+1

Hence, xy is odd

Let x be the odd integer

Difference of two squares is

m^2-n^2=(m+n)(m-n)

Let, m+n=x,m-n=1

So we get

m=(x+1)/2,n=(x-1)/2

We can see ,m and n will be integers because x is odd.

HEnce, proved.