Question

if A and B are two odd positive integers such that A is greater than B then prove that one of the two numbers A + B by 2 and A minus b by 2 is odd and the other is event ​

Answer 1

Answer:

One of the integers $\frac{a + b}{2}$ and $\frac{a - b}{2}$ is even and other is odd

Step-by-step explanation:

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

∴ (2m +1) > (2n + 1)

⇒ 2m > 2n

⇒ m > n or m - n > 0

∴ $\frac{a + b}{2}$ > 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 → (1)

It is already proved that $\frac{a + b}{2}$ and $\frac{a - b}{2}$ are positive integers → (2)

Recall that the difference between an odd number and even number is always an odd number.

Hence from (1) and (2), we can conclude that one of the integers $\frac{a + b}{2}$ and $\frac{a - b}{2}$ is even and other is odd.