Question

If a and b are two odd positive integers such that a > b, then prove that one of the two numbers `(a+b)/2` and `(a-b)/2` is odd and the other is even.

Answer 1

We know that any odd positive integer is of the form 4q+1 or, 4q+3 for some whole number q.

Now that its given a > b

So, we can choose a = 4q + 3 and b = 4q + 1

∴ `((a + b))/2 = [(4q + 3)+(4q + 1)]/2`

=> `((a + b))/2 = ((8q + 4))/2`

=> `((a + b))/2`

= 4q + 2 = 2(2q + 1) which is clearly an even number.

Now, doing `((a - b))/2`

=> `((a - b))/2 = [(4q + 3)-(4q + 1)]/2`

=> `((a - b))/2 = ((4q + 3 - 4q - 1))/2`

=> `((a - b))/2 = ((2))/2`

=> `((a - b))/2 = 1` which is an odd number.

Hence, one of the two numbers `((a + b))/2` and `((a - b))/2` is odd and the other is even.