If a and b are positive odd integers, then prove that one of (a+b)/2 and (a-b)/2 are even or odd number. I mean in the above numbers one is odd and one is an even number.

Let a>b

a=2n+1+2k ,n,k is positive
b=2n+1
(a+b)/2=2n+1+k
(a-b)/2=k
If k is odd,
then
(a+b)/2 is even
(a-b)/2 is odd

If k is even,
(a+b)/2 is odd
(a-b)/2 is even

The same method also applies if a<b
(However I assume that a,b are distinct number because if a=b, (a-b)/2=0 but 0 is neither odd nor even)

excuse me? 0 is an even number. The definition of even is that it leaves a zero remainder when divided by 2.

sorry about that....

I thought 0 is neither even nor odd....

To prove that one of (a+b)/2 and (a-b)/2 is odd and the other is even, we can look at the parity properties of odd and even numbers.

First, let's consider the sum of two odd numbers. When you add two odd numbers, the result is always even. This is because an odd number can be represented as (2n+1) for some integer n. So, for two odd numbers, we have:

(a + b) = (2m + 1) + (2n + 1) = 2(m + n + 1)

Notice that (m + n + 1) is an integer, let's call it k. The sum (a + b) can be written as 2k, which is even.

Next, let's consider the difference of two odd numbers. When you subtract one odd number from another odd number, the result can be either even or odd. This is because subtracting (2n + 1) from (2m + 1), where m and n are integers, can give us two different cases:

Case 1: If m > n, then (2m + 1) - (2n + 1) = 2(m - n)
The difference (a - b) can be written as 2k, which is even.

Case 2: If n > m, then (2n + 1) - (2m + 1) = 2(n - m)
The difference (a - b) can be written as 2k, which is even.

In both cases, the difference (a - b) can be expressed as 2k, where k is an integer. Therefore, (a - b) can be either even or odd.

Now, let's analyze the expressions (a+b)/2 and (a-b)/2:

For (a + b)/2, we know that (a + b) is even. When we divide an even number by 2, the result is always even. Therefore, (a + b)/2 is even.

For (a - b)/2, we have shown that (a - b) can be even or odd. When we divide an even number by 2, the result is always even. When we divide an odd number by 2, the result is always odd. Therefore, (a - b)/2 can be either even or odd.

In conclusion, we have proved that one of (a + b)/2 and (a - b)/2 is even, and the other can be either even or odd depending on the value of (a - b).