Question

show that every positive integer is neither even or odd.?

Answer 1

Let I be a positive integer.
We can now write I in the form of 
I = pq + r
So, pq + r can either be even or odd, 

Answer 2

Every positive intiger is either odd or even...

(if we divide any positive intiger by 2 either it is fully divisible by 2 or not
means either even or odd)

it is not possible that ever positive intiger is neither odd or even