Advertisement Remove all ads

Show that Every Positive Integer is Either Even Or Odd? - Mathematics

Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads

Show that every positive integer is either even or odd?

Advertisement Remove all ads

Solution

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.

Concept: Euclid’s Division Lemma
  Is there an error in this question or solution?

APPEARS IN

Video TutorialsVIEW ALL [1]

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×