Question

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

Answer 1

Hint: In this question knowledge of even and odd numbers is must and also remember general representation of odd numbers is given as; 2m + 1 where n is any whole number, use this information to approach the solution.

Complete step-by-step answer:

Let “a” and “b” are any two odd positive integers.

In general form it can be written as $a = 2m + 3\;$ and $\;b = 2m + 1$. Where m is any positive integers. 

And we know that $a > b$.

i.e. $2m+3 > 2m+1$

Consider, $\;\dfrac{{a + b}}{2} = \dfrac{{(2m + 3\;) + (2m + 1)}}{2} = 2m+2 = 2(m+1)$

Here the result i.e. 2(m+1) is an even number.

Therefore, $\dfrac{{a + b}}{2}$ Is an even number.

Now, $\dfrac{{a - b}}{2} = \dfrac{{(2m + 3\;) - (2m + 1)}}{2} = \dfrac{2}{2} = 1$

And we know that 1 is an odd number.

Hence, $\dfrac{{a - b}}{2}$ is an odd number.

From the above explanation we can conclude that one of the integers $\dfrac{{a + b}}{2}$ and $\dfrac{{a - b}}{2}$ is even and other is odd.

Note: The number which is divisible by 2 is known as an even number. And the number which is not divisible by 2 is known as an odd number. Sum of two even numbers is always an even number and the sum of two odd numbers is also an even number. But the sum of an even and odd number is always an odd number.