Question

Prove that out of any two consecutive positive integers one and only one is even

Answer 1

First i will define what is even number and what is odd number

Even number is the number which is divisible by 2 and even number if of form 2n where n can be any integer value,and difference between any 2 consecutive number will be 2

Odd number is the number which is not divisible by 2 and odd number is of form 2n+1 where n can be any integer value and difference between any 2 consecutive odd number will be 2

Let N and N+1 are two consecutive number

Assume both numbers are even

Than N/2 and (N+1)/2 will be completely divisible by 2

(N+1)/2 = (N/2) + 1/2

Since N/2 is also even number it will give zero remainer but when we divide 1 by 2 we will get 1 as remainder so remainder will be 1

So N+1 can not be even number because it is not divisible by 2 it is giving remainder as 1

So both consecutive number can not be even number

Similarly we can prove both consecutive number can not be odd number

Hence we can say two consecutive integer has only one number as even number

Example 3 and 4 in this only 4 is even number

Example 6,7 in this only 6 is even number