kvcpk wrote:selango wrote: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 positive distinct factors of n is odd.
(1)n has exactly two positive factors.
Let n= 3, n has only 2 positive factors ->1,3
Let n=2, n has exactly 2 positive factors -> 1,2
INSUFF
(2)The difference of any two positive distinct factors of n is odd.
A Positive integer which has exactly two positive factors is a prime number.
One of the factors is 1. For the difference to be odd, all other factors should be even.
But difference between the even factors will turn out to be odd. Hence only 1 even factor should exist.
2 is the only Even prime.
Hence n= 2.
SUFF
pick B
Whats OA?
you got the right answer, but this is an incorrect analysis of statement (2).
specifically, you wrote "a positive integer which has exactly 2 positive factors" --> this means that you are carrying over that condition from statement 1. the problem is that we don't know anymore, in this statement, that n only has two positive factors!
(in other words, you are actually analyzing the two statements
together when you think you are analyzing statement 2 by itself.)
here's the way it works:
imagine the number n = 2; this satisfies statement (2). the only factors are 1 and 2, so the only difference is 2 - 1 = 1, which is odd.
NOW, let's say that we add ANY other factor. let's call it "k".
here's the problem:
if the new factor "k" is
odd, then (k - 1) will be even, and so statement (2) won't be true anymore.
if the new factor "k" is
even, then (k - 2) will be even, and so statement (2) won't be true anymore.
the above proves that we can't add in
any other factor, and so n = 2 must be the only number that satisfies this statement.
--
incidentally, this is a very old problem, and i think that the gmat is moving gradually away from this sort of thing -- it's a very theory-heavy type of problem, similar to the type of reasoning used in very formal mathematical proofs.
from the new problems that i've seen lately from gmac, i would say that they're gradually moving away from these sorts of overly theoretical problems, and toward problems that are more mathematically lowbrow, but perhaps a little "trickier" (such as, e.g., sets of 2 simultaneous equations in 2 variables, but in which you only need 1 of the equations to solve).
