Question

show that every positive integer is either even or odd

Answer 1

Step-by-step explanation:

let us assume that there exist a small positive integer that is neither odd or even, say n.

Since n is least positive integer which is neither even nor odd, n - 1 must be either or or even.

CASE 1 :

If n - 1 is even , then n - 1 = 2m for some integer m .

But , => n = 2m + 1 .

This implies n is odd .

CASE 2 :

If n - 1 is odd , then n - 1 = 2m + 1 for some integer m .

But, => n = 2m + 2 = 2( m + 1 ) .

This implies n is even .

In both cases , there is a contradiction .

Thus , every positive integer is either even or odd .

Hence, it is solved

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