Question:

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.

The answer here is B.


I had selected answer choice C.

I was able to understand that (1) --> n = prime

From (2) I thought that since we know E-O = O we need an even number and an odd number.. but I did not see how this was enough information.

When I combined (1) and (2) I thought that the only way to chose an even and an off prime is for one of those primes to be 2. This is how I conculded that n must equal 2.

I wonder if someone can help clarify this question. I found the solution in the OG very difficult to follow..

From the question stem we already know that n is an integer and it is greater than 1. We want to know if n is equal to 2. The OG describes two cases where n> 2 that are impossible.. and then concludes that n must therefore = 2. I just wonder if someone follows the cases that are explained in the OG...

Thanks,
Carla

Carla,

Let me try to explain.

Statement II reads: The difference of any two distinct positive factors of n is odd.

This statement tells us that n is a prime number. "Two distinct" positive factors means a number is prime. For example, the number 3 (which is prime) has two distinct postive factors. The number 6 (which is not prime) has three distinct postive factors: 3, 2, and 1.

As 2 is the only prime number whose factors are Odd and Even, 2 is the only number n can be.

Hope that helps!

Hi,

Thanks again for helping me! I had not realized that "two distinct" meant "exactly two" factors which of course means prime. Have you seen other examples where this wording is used to indicate primeness? Once we know that it is prime the rest was clear for me as far as 2 being the only even prime etc..

Carla

Carla,

Just to clarify, I don't think "two distinct" means "exactly two" - I think my earlier post might have alluded to that, however, I just re-read what I wrote, so allow me to clarify what I meant.

6 has 3 distinct factors (3,2,1) - the difference between any two of these three factors is not always odd. For example, 3-2 = odd; 2-1 = odd; 3-1 = even. So, 6 cannot be n as Statement II states "The difference of any two distinct positive factors of n is odd.

I hope that helps!

By the way, I think this is quite a difficult question! I know I got it wrong the first time I attempted it, choosing C as well! :)

Quick citation note: This is from the Official Guide for GMAT Review, 11th edition. For copyright reasons, we must cite full source name.

Great discussion, I think this will be useful to a lot of people. In case it helps, I generally read "distinct" as "different." So "two distinct positive factors" = "two different positive factors" as GMAT 5/18 illustrated.