If a and b are two odd positive integers such that a b, then prove that one of the two numbers and is odd and the other is even.

Question

If a and b are two odd positive integers such that a > b, then prove that one of the two numbers and is odd and the other is even.

Philoid · Accepted Answer

Let the one number be 2p + 1

Let the other number be 2q + 1

As per question: p>q

As per first condition of the question: ⇒ ⇒ p+q+1

As per second condition of the question: ⇒ ⇒ p-q

Since adding two even numbers gives an even number and adding two odd numbers gives an even number then the two numbers cannot both be even or both odd.
Therefore one must be odd and one even.

EXAMPLE:

Take a = 7 and b = 3 such that a>b

Now, = = 5 which is odd

And = = 2 which is even