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Source: — Data Sufficiency |
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by GMATGuruNY » Thu Oct 10, 2013 5:57 pm
Check my post here:

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by theCodeToGMAT » Thu Oct 10, 2013 6:11 pm
PLEASE USE SPOILER TO HIDE THE ANSWERS

x --> INT

Statement 1:
x > 3!
3! = 6
So, X can be "7" , "8", etc.
7 --> doesn't have factor greater than "1"
8 --> has factor greater then "1" i.e. "2" & "4"
INSUFFICIENT


Statement 2:
15! + 2 ≤ x ≤ 15! + 15
15! => 1 x 2 x 3 ........ x 15
1 x 2 x 3 ........ x 15 + 2 ≤ x ≤ 1 x 2 x 3 ........ x 15 + 15
2(1x3......15)+2 ≤ x ≤ 15(1x2x3...14) + 15
2((1x3..15)+1) ≤ x ≤ 15((1x2x3..14)+1)
Similarly, for each factor of "X" between this range,, we can take out one "common" integer.
So, factor exists.
SUFFICIENT

Answer [spoiler]{B}[/spoiler]
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by Brent@GMATPrepNow » Thu Oct 10, 2013 6:42 pm
[email protected] wrote:If x is an integer, does x have a factor n such that 1 < n < x?

(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
This is a great candidate for rephrasing the target question.
What kinds of integers have a factor n such that 1 < n < x? Non-prime integers. So, we're really just asking whether or not n is a non-prime integer. Let's make it even easier on ourselves and ask . . .

Rephrased target question: Is x prime?

Statement 1: x > 3!
In other words, x > 6
case a) x = 7, in which case x is prime
case b) x = 8, in which case x is not prime
Statement 1 is NOT SUFFICIENT

Statement 2: 15! + 2 ≤ x ≤ 15! + 15
This is saying that x can have one of 14 different possible values. So, let's begin checking some values.

Is 15! + 2 prime? No.
Notice that 15! = (15)(14)(13)...(3)(2)(1)
So, we can factor a 2 out of 15! + 2, to get: 15! + 2 = 2[(15)(14)(13)...(3)(1) + 1]
This means that 2 is a factor of 15! + 2, which means it is not prime.

Next, 15! + 3 prime? No.
Notice that 15! = (15)(14)(13)...(4)(3)(2)(1)
So, we can factor a 3 out of 15! + 3, to get: 15! + 3 = 3[(15)(14)(13)...(4)(2)(1) + 1]
This means that 3 is a factor of 15! + 3, which means it is not prime.

We can continue this process to show that none of the 14 possible values of x are prime.
As such, statement 2 is SUFFICIENT and the answer is B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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