Show that every positive integer is either even or odd.

Let us assume that there exist a smallest positive integer that is neither odd nor even, say n. Since n is 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 (k+1) this implies n is even.

In both ways we have a contradiction.

Thus, every positive integer is either even or odd

Was this answer helpful?

  
   

3.5 (20)

Upvote (17)

Choose An Option That Best Describes Your Problem

Thank you. Your Feedback will Help us Serve you better.

Leave a Comment

Your Mobile number and Email id will not be published. Required fields are marked *

*

*

BOOK

Free Class

Ask
Question