prime number

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prime number

by clock60 » Thu Nov 25, 2010 1:53 pm
hi guys. i spend too much time solving that problem. can somebody provide any shortcut

If x is a positive integer, is x! + (x + 1) a prime number?
(1)x < 10
(2) x is even

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by Rahul@gurome » Thu Nov 25, 2010 7:23 pm
First consider (1) alone.
x<10.
For x = 1, 2, 3, 4 the values of x! + (x + 1) are 2, 5, 10, 29 respectively.
Out of these, 3 are prime and 1 is not prime.
Or (1) alone is not sufficient.
Next consider (2) alone.
For x = 2, 4, 6, 8 the values of x! + (x + 1) are 5, 29, 727, 8!+9.
The first 3 are prime but 8! + 9 will not be a prime because it will be divisible by 9.
Again nothing definite can be said.
So even (2) alone is not sufficient.
On combining we can check again with same values of 2, 4, 6 and 8 and get the same result that nothing definite can be said .
The correct answer is E.
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by Laura GMAT Tutor » Fri Nov 26, 2010 11:37 am
Rahul's explanation is great. :) But I wanted to throw in something else...

He said that "8! + 9" will be divisible by 9, and I just wanted to clarify why that was, in case anyone was wondering.

First, of course, 8! is (8)(7)(6)(5)(4)(3)(2)(1). Will 8! be a multiple of 9? Absolutely! It has a 6 and a 3, meaning that it has two 3s in its prime factor list. Since 8! must be a multiple of 9, 8!+9 must also be a multiple of 9. So it can't be prime.
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by rishab1988 » Fri Nov 26, 2010 12:32 pm
great explanation guys