Question

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

Answer 1

Let a and b are any two odd positive integers.
Hence a=2m+1 and b=2n+1 where m and n are whole numbers.
Consider $\frac{a+b}{2}=\frac{(2m+1)+(2n+1)}{2}=\frac{2m+2n+2}{2}=(m+n+1)$
Therefore $\frac{a+b}{2}$ is a positive integer.
Now, $\frac{a+b}{2}=\frac{(2m+1)-(2n+1)}{2}=\frac{2m-2n}{2}=(m-n)$
But given $a>b$
$\therefore (2m+1)>(2n+1)$
$\Rightarrow 2m>2n$
\Rightarrow m> n or m -n > 0
$\therefore \frac{a-b}{2}>0$
Hence $\frac{a-b}{2}$ is also a positive integer
Now we have to prove that of the numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and another is even number.
Consider, $\frac{a+b}{2}-\frac{a-b}{2}$
$=\frac{a+b-a+b}{2}=\frac{2b}{2}=b$ which is an odd positive integer