Show that every positive integer is even or odd
Asked by admin @ in Math viewed by 283 People
Show that every positive integer is either even or odd.
Answered by admin @
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.
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