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If x is a positive integer greater than 1, is x! + x + 1 a p

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If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 08 Jul 2014, 07:07
1
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Difficulty:

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Question Stats:

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If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10

(2) x is odd

OE
(1) INSUFFICIENT
If x = 2: 2! + (2 + 1) = 5, which is prime.
If x = 3: 3! + (3 + 1) = 6 + (3 + 1) = 10, which is not prime.

(2) SUFFICIENT:
If x = 3: 3! + (3 + 1) = 10, which is not prime.
If x = 5: 5! + (5 + 1) = (5 × 4 × 3 × 2 × 1) + 6.
This expression must be divisible by 3, since both of its terms are divisible by 3.
Furthermore it must be even, because both terms are even.
Therefore, it is not a prime number.
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 08 Jul 2014, 07:55
2
5
goodyear2013 wrote:
If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10

(2) x is odd

OE
(1) INSUFFICIENT
If x = 2: 2! + (2 + 1) = 5, which is prime.
If x = 3: 3! + (3 + 1) = 6 + (3 + 1) = 10, which is not prime.

(2) SUFFICIENT:
If x = 3: 3! + (3 + 1) = 10, which is not prime.
If x = 5: 5! + (5 + 1) = (5 × 4 × 3 × 2 × 1) + 6.
This expression must be divisible by 3, since both of its terms are divisible by 3.
Furthermore it must be even, because both terms are even.
Therefore, it is not a prime number.


If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10.

If x=2, then x! + x + 1 = 5 = prime.
If x=3, then x! + x + 1 = 10, not a prime.

Not sufficient.

(2) x is odd. So, x is an odd integer greater than 1: 3, 5, 7, ... In this case x! + x + 1 = even + odd + odd = even > 2, not a prime. Sufficient.

Answer: B.
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 09 Jul 2014, 01:19
Let X<10, then
for the x=2
x! + x + 1 = 5 = prime number
& for the x=3
x! + x + 1 = 10, not a prime number
So it is not possible if x is a positive integer greater than 1, is x! + x + 1 a prime number..
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 01 Oct 2014, 06:27
Bunuel wrote:
goodyear2013 wrote:
If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10

(2) x is odd

OE
(1) INSUFFICIENT
If x = 2: 2! + (2 + 1) = 5, which is prime.
If x = 3: 3! + (3 + 1) = 6 + (3 + 1) = 10, which is not prime.

(2) SUFFICIENT:
If x = 3: 3! + (3 + 1) = 10, which is not prime.
If x = 5: 5! + (5 + 1) = (5 × 4 × 3 × 2 × 1) + 6.
This expression must be divisible by 3, since both of its terms are divisible by 3.
Furthermore it must be even, because both terms are even.
Therefore, it is not a prime number.


If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10.

If x=2, then x! + x + 1 = 5 = prime.
If x=3, then x! + x + 1 = 10, not a prime.

Not sufficient.

(2) x is odd. So, x is an odd integer greater than 1: 3, 5, 7, ... In this case x! + x + 1 = even + odd + odd = even > 2, not a prime. Sufficient.

Answer: B.



Is there any other way , apart from testing number. I solved it correctly but took more than 2 min.
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 13 Apr 2016, 07:45
(1) x < 10.

If x=2, then f(x)= 5 => prime
If x=3, then f(x) = 10 => not prime

Not suff.

(2) x is odd.

x! + x + 1 = x(x-1)(x-2)(x-3)...(x-n) + x + 1 (with n always is even)

=> x[1 + (x-1)(x-2)(x-3)...(x-n)] + 1

We have (x-1) always even, (x-2) odd, (x-3) even, ... (x-n)odd => (x-1)(x-2)(x-3)...(x-n) = even

=> [1 + (x-1)(x-2)(x-3)...(x-n)] = odd (1+ even)
=> x[1 + (x-1)(x-2)(x-3)...(x-n)] = odd (odd * odd)
=> x[1 + (x-1)(x-2)(x-3)...(x-n)] + 1 = even (odd + 1)

=> suff.

I just write in detail, it would be very quick to infer that.
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 29 Sep 2016, 06:47
1
goodyear2013 wrote:
If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10

(2) x is odd



I got to B in ~1 minute..
1. x<10 -> 1<x<10
suppose x=2.
2+2+1 = 5
suppose x=3
3! = 6.
6+3+1 = 10 - not prime.

2 outcomes - insufficient.

2. x is odd.
since x can't be 1, x! of any odd number will always be even.
since x is odd, x+1 will always be even
now..x!(even) + x+1(even) = even.
even number can't be a prime number.
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 08 Apr 2017, 04:49
definitely no. this is why option b ?

please help anyone :roll: :roll: :wall
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If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 08 Apr 2017, 04:59
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 08 Apr 2017, 05:08
thank bunuel sir !

:-D :) :-D :)
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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New post 28 May 2017, 16:32
him1985 wrote:
Bunuel wrote:
goodyear2013 wrote:
If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10

(2) x is odd

OE
(1) INSUFFICIENT
If x = 2: 2! + (2 + 1) = 5, which is prime.
If x = 3: 3! + (3 + 1) = 6 + (3 + 1) = 10, which is not prime.

(2) SUFFICIENT:
If x = 3: 3! + (3 + 1) = 10, which is not prime.
If x = 5: 5! + (5 + 1) = (5 × 4 × 3 × 2 × 1) + 6.
This expression must be divisible by 3, since both of its terms are divisible by 3.
Furthermore it must be even, because both terms are even.
Therefore, it is not a prime number.


If x is a positive integer greater than 1, is x! + x + 1 a prime number?

(1) x < 10.

If x=2, then x! + x + 1 = 5 = prime.
If x=3, then x! + x + 1 = 10, not a prime.

Not sufficient.

(2) x is odd. So, x is an odd integer greater than 1: 3, 5, 7, ... In this case x! + x + 1 = even + odd + odd = even > 2, not a prime. Sufficient.

Answer: B.



Is there any other way , apart from testing number. I solved it correctly but took more than 2 min.


The main takeaways from this question is this: 2 is the ONLY number (apart from 1) that, when expressed as a factorial, will give you a prime # (when you add "x" and "1"). Why, you ask? B/c any number > 2 will include "2" in its prime factorization. Meaning, if you have 3! = 3x2; or 5! = 5x4x3x2.

ALSO, any number multiplied by an even will be EVEN. and the ONLY EVEN PRIME = 2.

Therefore, (1) = insufficient b/c x<10 includes 2 (prime) as well as 3, 4, 5, 6, 7, 8, 9 (which are all NOT prime).
(2) is sufficient b/c we know "1" cannot be included here, so the only numbers to play with are: 3, 5, 7, 9 -- which are all NOT prime. This is sufficient b/c we can safely say "x! + x + 1 is NOT PRIME" under these conditions
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Re: If x is a positive integer greater than 1, is x! + x + 1 a p  [#permalink]

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Re: If x is a positive integer greater than 1, is x! + x + 1 a p   [#permalink] 23 Dec 2018, 10:34
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