1 has exactly 1 factor : 1
But it is already mentioned that X > 1
So, X cannot be equal to 1.

Now, 2 has exactly 2 factors : 1 and 2
So, X can be equal to 2.

Now, for any integer X greater than 2, the number of unique factors of X must be less than X.

Why so?
Because any X greater than 2 will not be divisible by (X - 1)

Why so?
Because X and (X - 1) are consecutive integers.
So, one of them will be odd and the other one will be even.
So, (X - 1) cannot divide X.

So, only possible value of X is 2.

Hope that helps.

Not understood.
Now, for any integer X greater than 2, the number of unique factors of X must be less than X.
This above statement is understood by me by the fact that factors cant be greater than the number.

Why so?
Because any X greater than 2 will not be divisible by (X - 1)? Why this statement has come?
This X-1 is if purely notation for consecutive integer then why not X+1 and as a fact any number which is consecutive can not be divisible by its immediate before number. Right?

Why so?
Because X and (X - 1) are consecutive integers.
So, one of them will be odd and the other one will be even.
So, (X - 1) cannot divide X. Why this statement?

For any positive integer x to have x unique factors means that it must be a multiple of every integer from 1 to x.
So, x must be a factor of x-1, which can't be true for any x > 2.
So, x = 2 is the only solution.

I got your point!!!thanks for the info.!!!you must be a expert in maths...have you taken gmat, if yes kindly tell me which book to refer after manhattan gmat maths.
One more thing is why not X+1 is talked about?

Anuj