Question

show that every positive integer is either even or odd​

Answer 1

Let n is a positive integer . The basic principle is " when positive n is either odd or even then (n + 1) is also either even or odd .

Means if n is odd then (n +1) should be even and if n is even then (n+1) should odd.

Case 1 :- when n is odd e.g., n = 2k + 1 , where k is integer then, (n +1) = (2k+1)+ 1

= (2k +2) , divisible by 2 hence, (n +1) is even .

Case 2:- when n is even e.g., n = 2k , where k is integer then (n +1) = 2k +1

doesn't divisible by 2 , so, (n +1) is odd integer .

From case1 and case2 it is clear that if n is positive then it is either odd or even.

Answer 2

Let us assume that there exists a smallest positive integer that is neither odd nor even, say n. Sincen is the least positive integer which is neither even

nor odd, n - 1 must be either odd or even.

Case 1: If n - 1 is even, n - 1 = 2k for some k. But this implies n 2k + 1

this implies n is odd.

Case 2: If n - 1 is odd, n 1 2k + 1 for some k. But this implies n = 2k + 2 = 2(k + 1) this implies n is even.

In both ways we have a contradiction. Thus, every positive integer is either even or odd.