VeritasPrepKarishma wrote:

If you have done a variety of questions, you have come across this concept somewhere before. The point is to recognize which concept I am talking about.

Ques. Given that x and y are positive integers, is x prime?

I. \((y + 1)! <= x <= (y + 1)(y! + 1)\)

II. \((y + 1)! + 1\) has five factors

Let's analyze the question.

The question stem just tells us that x and y are positive integers. The information was provided mainly to rule out a decimal value for x and since we are using factorials for y.

Question: Is x prime?

I. \((y + 1)! <= x <= (y + 1)(y! + 1)\)

This needs to be modified a little. Why? because right now, there is no apparent relation between (y+1)! and (y + 1)(y! + 1). (y+1)! makes sense to me, (y! +1) does not. Can I bring everything in terms of (y+1)?

I see that I have a (y+1) multiplied with y! and with 1. Recognize that (y+1)*y! = (y+1)!

So, (y + 1)(y! + 1) = (y+1)y! + (y+1) = (y+1)! + (y+1)

How will you know that this is how you would like to split it? Use (y+1)! on left side as a clue. The right side should make sense with respect to the left side which is in its simplest form.

\((y + 1)! <= x <= (y+1)! + (y+1)\)

Now think about it. x can take any of the following values (general case):

(y+1)! - Not prime except if y is 1

(y+1)! + 1 - Cannot say whether it is prime or not. If y = 1, this is prime. If y is 2, this is prime. If y is 3 it is not.

(y+1)! + 2 - Has 2 as a factor. Not prime

(y+1)! + 3 - Has 3 as a factor. Not prime

.

.

(y+1)! + (y+1) - Has (y+1) as a factor. Not prime

Hence x may or may not be prime.

II. \((y + 1)! + 1\) has five factors

Not sufficient on its own. No mention of x.

Together: There were two exceptions we found above.

1. If y = 1, then (y+1)! is prime.

From stmnt II, since (1+1)! + 1 = 3 does not have 5 factors (It only has 2), y is not 1. Hence y is not 1 and (y+1)! is not prime

2. We don't know whether (y+1)! + 1 is prime

Since (y+1)! + 1 has 5 factors, it is definitely not prime. Prime numbers have only 2 factors.

Since both exceptions have been dealt with using both statements together, we can say that in every case, x is not prime.

Answer (C).

The question is not time consuming if you quickly see the pattern. It's all about getting exposed to the various concepts. The explanation seems long but only because I have written out everything my mind thinks in a few seconds. This is a relatively tough question but not beyond GMAT. Expect such questions if you are shooting for 49-51 in Quant.

_________________

Karishma

Veritas Prep | GMAT Instructor

My Blog

Get started with Veritas Prep GMAT On Demand for $199

Veritas Prep Reviews