suryapal wrote:but statement second says that difference of any distinct factor
so if we take 6,1,2,3 and 6 are factors of 6 and 6-3=3 , this satisfy the given condition , right???
hence statement second alone is not sufficient.. if i'm wrong .. please explain
Read the second option a bit more carefully ...
(2) the difference of any two distinct positive factors of n is odd
It means you have to consider all the pairs that you can make out of its factors.
Take for e.g 6, the # that you took. You are correct in saying that 1,2,3 and 6 are it's positive factors and difference of 6 and 3 is odd. But, what about difference of 3 and 1. Is it odd? The condition should satisfy for ANY two factors of the number.
and that is why 2 is sufficient :
* consider any prime greater than 2. It will have 1 and the number itself as the only factors. But, the difference of the two factors (only pair) will be even.
* Consider any other odd composite #. it will have atleast two odd factors and hence, difference of these two factors will be even.
*any even # greater than two. It will also have 2 and that number as its factor and hence, won't satisfy the condition.
NOW, consider 2.
its factors 1 & 2 . We can make just a single pair out of these two factors and their difference is ODD. which is what we want to know.
Therefore, second statement is sufficient to tell that the number is 2.
