All odd numbers have an even number on either side of them. All even numbers are divisible by 2. So all odd numbers can be represented by the expression 2x+1 where x is any integer. Let another odd number be 2y+1.
Now subtract one from the other: 2(x-y), the ones cancel out. No matter what x and y are, their difference is doubled, making it even, therefore the difference between two odd numbers is always even.
