Question

If a and bare two odd positive integers such that a > b, then prove that one of the two numbers
a+b/2 and a-b/2
is odd and the other is even.
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Answer 1

let a=2n+1 and b=2m+1,

where m and n are positive integers

then a+b/2=m+n+1

and a-b=n-m

now suppose that m and n both are odd, then n-m=even and m+n+1= odd

similarly take all the cases: m even, n odd ; n even, m odd ; or both even

you'll always get one of them to be even and other to be odd