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

First we can easily verify that a+b/2 and a-b/2 are positive integers since the sum of two odd numbers is always even and, the difference of two odd numbers is always even respectively.

This implies that on division by 2 we will have a positive integer.

Let

x= a+b/2 + a-b/2

therefore x=a

Therefore, we have that x is an odd positive integer. We know that the sum of two even or sum of two odd numbers is never odd. Thus, it follows that a+b/2 is even when a-b/2 is odd and vice-versa.

Hence proved.

hope it helps