Question

If a and b are 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 .

Answer 1

Let, the two odd numbers are given by:

$a=2m+1$

And $b=2n+1$

Where, $m,n\in Z$

So, $\frac{a+b}{2}=\frac{2m+1+2n+1}{2}$

$=\frac{2(m+n+1)}{2}$

$=m+n+1$

And,

$\frac{a-b}{2}=\frac{2m+1-(2n+1)}{2}$

$=\frac{2(m-n)}{2}$

$m-n$

Now, for any values of $mandn$, If $m+n$ is odd, $m-n$ is also odd and vice-versa.

So, if $=m+n+1$ is odd, $m-n$ is even or vice-versa.

Hence, one of the two numbers a+b/2 and a-b/2 is odd and the other is even.