Question

EXERCISE 1.1
1. If a and b are two odd positive integers such that a > b, then prove that one of the two
ab a-b
numbers and is odd and the other is even.
22
را دارد​

Answer 1

Answer:

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 the 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.