# 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

### Asked by shabnakader | 22nd Jun, 2017, 06:24: PM

Expert Answer:

### 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.

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.

### Answered by Rebecca | 22nd Jun, 2017, 07:22: PM

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