If a and b are positive odd integers, then prove that one of (a+b)/2 and (a-b)/2 are even or odd number. I mean in the above numbers one is odd and one is an even number.

Answer 1

Let a>b
a=2n+1+2k ,n,k is positive
b=2n+1
(a+b)/2=2n+1+k
(a-b)/2=k
If k is odd,
then
(a+b)/2 is even
(a-b)/2 is odd

If k is even,
(a+b)/2 is odd
(a-b)/2 is even

The same method also applies if a<b
(However I assume that a,b are distinct number because if a=b, (a-b)/2=0 but 0 is neither odd nor even)

Answer 2

excuse me? 0 is an even number. The definition of even is that it leaves a zero remainder when divided by 2.

Answer 3

sorry about that....
I thought 0 is neither even nor odd....