Question

Show that every positive integer is either even or odd.

Answer 1

Hint:- Let ‘n’ be a positive integer. The basic concept is that "when a positive integer ‘n’ is either odd or even then ‘(n + 1)’ is also either even or odd.

Complete step-by-step answer:

For the solution of this question, we are going to take 2 examples or cases:-

Case 1:- When ‘n’ is an odd number. For example, ‘n = 2k + 1’, where ‘k’ is an integer,
then, (n +1) = (2k+1) + 1. And after simplifying the brackets, we get (2k +2), which is divisible by 2. And as (2k+2) is divisible by 2, it must be an even number, because only even numbers are a multiple of 2. Hence, (n +1) is also even.

Case 2:- When ‘n’ is an even number. For example, ‘n = 2k’, where ‘k’ is an integer,
then, (n +1) = 2k +1. And as we can see that (2k+1) is not divisible by 2, therefore (2k+1) is an odd number. Hence, (n +1) is also an odd integer.
From case 1 and case 2 described above it is clear that if n is positive then it is either odd or even.

Note:-
The student can make an error by not representing an odd number as (2k+1) and even number as (2k). In other words, if the student has taken (2k+1) as an even number and (2k) as an odd number, then there will be an error in the solution.