Question

show that every positive integer is either even or odd

Answer 1

We know that, INTEGERS can be further categorised into 3 types:

1. POSITIVE INTEGER.
2. NEGATIVE INTEGER.
3. ZERO INTEGER.

Positive integers are the natural numbers.

Negative integers are the negative of the natural numbers.

Zero (0) is the only zero integer as it can only be called as neither POSITIVE nor NEGATIVE.

So, taking positive integers into consideration(1,2,3......),we can say that:

They can be either ODD(1,3,5,7,9) or EVEN(2,4,6,8,10).

So,we can say that every positive integer is either ODD or EVEN.

Answer 2

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

THANKS

#BeBrainly.