Prove each of the following Show that if a is an odd number

Prove each of the following.

Show that if a is an odd number (i.e. of the form 2n + 1), then a^2 is an odd number. Show that if a is an even number (i.e. of the form 2n), that a^2 is an even number.

Solution

(1) Let a =(2n+1).(n, integer)

Then a² =(2n+1)(2n+1)

             =4n² +4n+1

            =4(n² +1) +1

             =2M+1, M an integer.

It follows that a² is an odd integer, if a is odd

(2) If a is an even number , then a=2n for some integer n.

Now a² =4n² =2(2n²) =2K, K an integer.

It follows that a² is an even integer., if a is even