Question

If a and b are two positive integer such that a>b the prove that one of the number a+b/2 and a-b/2 is odd and the other is even.

Answer 1

Solution:

Let a = 2q+3

b= 2q+1

Be two positive odd integers such that a>b

Now,

$\frac{a + b}{2} = \frac{2q + 3 + 2q + 1}{2} = \frac{4q + 4}{2} = 2q + 2 = \: an \: even \: number \:$

And

$= \frac{a - b}{2} = \frac{(2q + 3) - (2q + 1)}{2} = \frac{2q + 3 - 2q - 1}{2} = \frac{2}{2} = 1 = an \: odd \: number$

Hence one of the two numbers (a+b)/2 and (a-b)/2 is odd and the other is even for any two positive odd integer.