If a and b are two positive odd integers prove that one out of a+b÷2 and a-b÷2 is odd and other is even

Given that both a and b are positive odd integers.

Let a = 2j + 1 and b = 2k+ 1,  where j and k can be

i) both odd or

ii) both even or

iii) one odd and a > b

 

Now, if j and k are both odd and both even (cases i and ii) then,

Now j + k will be always even (sum of two odd number is even and sum of two even numbers is definitely even)

 

This means j + k + 1 is definitely odd  is definitely odd

 

and 

 

which will always be even since the difference of odd as well as even numbers is even.

 

Hence,  j - k is definitely even is definitely even.

 

Now for case (iii) lets say j is odd and k is even

Now j + k will be always odd (sum of one odd and one even number is always odd )

 

This means j + k + 1 is definitely even  is definitely even.

similar logic goes for  which will definitely be odd.

Hence proved.