akshatgupta87 wrote:Q.)Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.

Someone explain..

It can be useful to understand when a number will have an even number of divisors, and when a number will have an odd number of divisors. If you list all of the factors of, say, 12, they are all in pairs which give a product of 12:

1*12
2*6
3*4

So most numbers have an even number of factors. Perfect squares are an exception, however; they will always have one factor which is not in a 'pair'. Taking 36 for example:


1*36
2*18
3*12
4*9
6

We have an odd number of factors, because 6 is not in a pair.

The upshot is:
- perfect squares always have an odd number of factors
- if a number is not a perfect square, it has an even number of factors

If you know this, then you can see instantly that Statement 1 is sufficient here.

Statement 2 is also sufficient, but it tests something you'd never be expected to know for the GMAT. Since almost every test taker would find it difficult to establish that Statement 2 is sufficient within two minutes without resorting to number-picking, this is clearly a prep company question, and not a real GMAT question. I've explained why Statement 2 is sufficient in other posts, so you can probably find a solution with a forum search (this question has been posted here quite a few times) - you just need to establish that a perfect square has an odd number of odd divisors - but understanding why Statement 2 is sufficient is not going to help you on your GMAT, so I won't go into further detail here.