Question

If $a$ and $b$ are two odd positive integers such that $a>b$, then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even.

Answer 1

We have

$a$ and $b$ are two odd positive integers such that $a$ & $b$

but we know that odd numbers are in the form of $2n+1$ and $2n+3$ where $n$ is integer.

so, $a=2n+3,b=2n+1,n\in 1$

Given $\Rightarrow a>b$

now, According to given question

Case $I:$

$\frac{a+b}{2}=\frac{2n+3+2n+1}{2}$

$=\frac{4n+4}{2}$

$=2n+2=2(n+1)$

put let $m=2n+1$ then,

$\frac{a+b}{2}=2m\Rightarrow$ even number.

Case $II:$

$\frac{a-b}{2}=\frac{2n+3-2n-1}{2}$

$\frac{2}{2}=1\Rightarrow$ odd number.

Hence we can see that, one is odd and other is even.

This is required solutions.