Question

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

Answer 1

For $a,b$ being odd

$a+b=$even

and $a-b=$even

If $\frac{(a+b)}{2}=$odd

let $a=2p+1$

and $b=2q+1$

$⟹\frac{(a+b)}{2}=$odd$=p+q+1$

so, $p,q$ are even or odd both

So,$\frac{(a-b)}{2}=p-q$

Now, for $p,q$ to be even

$p-q=$even

and for $p,q$ to be odd

$p-q=$even.

So,$\frac{(a+b)}{2}$is odd and$\frac{(a-b)}{2}$is even.