Question

show that every positive integer is either even and odd

Answer 1

Let n be an arbitrary positive integer on dividing by two 'let m be quotient and r be remainder. Then n =2m+r,where o=<r <2 by eucluids division lemma . Therefore n=2m or n=2m+1 for some integer m. so n is either even or odd.

Answer 2

even and odd vary by one number meaningly
1 is odd
2 is even
3 is odd
4 is even
and so on