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x!+x+1

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j_shreyans Legendary Member Default Avatar
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x!+x+1

Post Mon Sep 08, 2014 9:56 pm
If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10

(2) x is odd

OAB

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Post Wed Sep 10, 2014 10:51 am
j_shreyans wrote:
If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10
(2) x is odd
Target question: Is (x! + x + 1) a prime number?

Given: x is a positive integer greater than 1

Statement 1: x < 10
There are several values of x that satisfy this condition. Here are two:
Case a: x = 2, in which case x! + x + 1 = 2! + 2 + 1 = 5. So, YES, x! + x + 1 IS prime
Case b: x = 3, in which case x! + x + 1 = 3! + 3 + 1 = 10. So, NOT, x! + x + 1 is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x is ODD
IMPORTANT: if x is an integer greater than 1, then x! must be EVEN
Notice that:
2! = (2)(1) = 2 (EVEN)
3! = (3)(2)(1) = 6 (EVEN)
4! = (4)(3)(2)(1) = 24 (EVEN)
5! = (5)(4)(3)(2)(1) = 120 (EVEN)
etc
-------
So, if x is ODD, then x! + x + 1 = EVEN + ODD + 1
= EVEN
If x! + x + 1 is even (and greater than 2), then x! + x + 1 definitely CANNOT be prime
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

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Top Reply
Post Mon Sep 08, 2014 11:36 pm
HI j_shreyans,

This question is perfect for TESTing VALUES.

We're told that X is a POSITIVE INTEGER greater than 1. We're asked if X! + X + 1 is prime. This is YES/NO question.

Fact 1: X < 10

If X = 2, then 2! + 2 + 1 = 5 and the answer to the question is YES.
If X = 3, then 3! + 3 + 1 = 10 and the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: X is odd

If X =3, then 3! + 3 + 1 = 10 and the answer to the question is NO.
If X = 5, then 5! + 5 + 1 = 126 and the answer to the question is NO.

From here, it's worthwhile to discuss a Number Property that applies….

Since X is Odd AND greater than1, X! will be EVEN (since the product includes at least 1 even number). Adding X (which is odd) + 1 (which is also odd), the sum MUST BE EVEN. The sum will be divisible by 2, so it will NOT be prime and the answer to the question will ALWAYS BE NO.
Fact 2 is SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

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