OG11 question 132 - properties of numbers

[GMATCHPOINT]

If the integer n is greater than 1, is n equal to 2?

(1) n has exactly two positive factors
(2) The difference of any two distinct positive factors of n is odd

Stmt 1 is clearly not sufficient. But can someone explain the stmt 2 (sufficient) better than OG11 pls?

[Stuart@KaplanGMAT]

If the integer n is greater than 1, is n equal to 2?

(1) n has exactly two positive factors
(2) The difference of any two distinct positive factors of n is odd

Stmt 1 is clearly not sufficient. But can someone explain the stmt 2 (sufficient) better than OG11 pls?

(1) n is prime - might be 2, might not.. insufficient.

(2) if the difference of any two distinct positive factors of n is odd, then n must have exactly 1 odd factor and 1 even factor. Only 2 satisfies this condition... sufficient.

Let's explore why:

If a number has at least two odd factors, then the difference between them will be even. Since all odd numbers have at least two odd factors, we can eliminate all odds from consideration.

If a number has at least two even factors, then the difference between them will be even. Since all even numbers greater than 2 have at least 2 even factors (i.e. 2 and the number itself), we can eliminate all evens greater than 2 from consideration.

Well, we know that n is an integer greater than 1. We've eliminated every odd integer; we've eliminated every even integer greater than 2. Therefore, n MUST be 2.

(2) is sufficient and (1) isn't: choose (B).

[GMATCHPOINT]

tks! now it is clear.