Question

Show that every positive integer is either even or odd.

Answer 1

Let $\mathrm{n}$ is a positive integer. The basic principle is that when any positive integer $\mathrm{n}$ is odd or even then $\mathrm{n}+1$is also either odd or even. It implies that if $\mathrm{n}$ is odd then $\mathrm{n}+1$ will be even and if $\mathrm{n}$ is even then $\mathrm{n}+1$will be

Two cases arises for the given question

Case 1: When $\mathrm{n}$ is odd

Step1:Let $\mathrm{n}=2\mathrm{k}+1$ where $\mathrm{k}$ is integer

Step2: $\mathrm{n}+1=2\mathrm{k}+1+1$

⇒ $\mathrm{n}+1=2\mathrm{k}+2$.

⇒$2\mathrm{k}+2=2(\mathrm{k}+2)$ is divisible by $2$.

Thus, $\mathrm{n}+1$ is even when $\mathrm{n}$ is odd.

Case 2: When $\mathrm{n}$ is even

Step1:Let $\mathrm{n}=2\mathrm{k}$ where $\mathrm{k}$ is integer

Step2: $\mathrm{n}+1=2\mathrm{k}+1$ .

⇒$\mathrm{n}+1=2\mathrm{k}+1$ is not divisible by $2$.

Thus, $\mathrm{n}+1$ is odd when $\mathrm{n}$ is even.

Based on the above two cases it is evident that every positive integer is either even or odd.