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# Is the positive integer x even?

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Is the positive integer x even?  [#permalink]

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28 Feb 2016, 12:11
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45% (medium)

Question Stats:

68% (01:47) correct 32% (01:59) wrong based on 237 sessions

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Is the positive integer x even?

(1) (x - 1) is a prime number
(2) (x^2 - 1) is a prime number

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Re: Is the positive integer x even?  [#permalink]

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28 Feb 2016, 12:32
1
Bunuel wrote:
Is the positive integer x even?

(1) (x - 1) is a prime number
(2) (x^2 - 1) is a prime number

Statement 1- the exception is 2 so can't say
Statement 2 - (x-1) (x+1) is odd so, both these terms should be odd (since they are alternate consecutive, they have to be the same type odd or even )and x would be even

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Re: Is the positive integer x even?  [#permalink]

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28 Feb 2016, 21:17
Bunuel wrote:
Is the positive integer x even?

(1) (x - 1) is a prime number
(2) (x^2 - 1) is a prime number

Statement 1: (x - 1) is a prime number
x = 3 implies (x - 1) = 2. A prime number
x = 4 implies (x - 1) = 3. A prime number.
Hence x can be both even and odd. Insufficient

Statement 2: (x^2 - 1) is a prime number
(x - 1)(x + 1) is a prime number.
The only possible value of (x - 1)(x + 1) can be 1*3 = 3
Therefore x = 2
Sufficient
Option B
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Re: Is the positive integer x even?  [#permalink]

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28 Feb 2016, 23:48
2
Bunuel wrote:
Is the positive integer x even?

(1) (x - 1) is a prime number
(2) (x^2 - 1) is a prime number

From the question stem, we get that:
$$x$$ is an integer and $$x>0$$
Now lets consider the statements:

Statement 1
$$(x - 1)$$ is a prime number
This gives us 2 scenarios:
either $$(x - 1)$$ is an even prime number (i.e. 2) or $$(x - 1)$$ is an odd primer number (i.e. any prime numbers except 2)
If $$(x - 1)$$ is an even prime number, then $$x = 3$$, an odd integer
If $$(x - 1)$$ is an odd prime number, then $$x$$ has to be an even integer
Therefore, we do not get a definitive yes/no from this statement. This statement is insufficient.
Hence, we can reject options A and D.

Statement 2
$$(x^2 - 1)$$ is a prime number.
This also gives us 2 scenarios:
either $$(x^2 - 1)$$ is an even prime number (i.e. 2) or $$(x^2 - 1)$$ is an odd primer number (i.e. any prime numbers except 2)
If $$(x^2 - 1) = 2$$; $$x^2 = 3$$. This is not possible as we know that $$x$$ is an integer.
Hence, the only possible scenario is that $$(x^2 - 1)$$ is an odd primer number.
This implies $$x^2$$ is an even number and accordingly, $$x$$ is an even number.
This statement is sufficient. We can reject options C and E.

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Re: Is the positive integer x even?  [#permalink]

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10 Aug 2016, 07:14
Excellent Question
Here in statement one prime number can be 2 so X can be even or odd
But in statement two x^2-1 can never be two as it would mean that x is not an integer which is contradictory
so x^2-1 must be odd prime => x^2= odd+odd=even so x must be even
Smash that B
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Re: Is the positive integer x even?  [#permalink]

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17 Dec 2018, 10:57
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Re: Is the positive integer x even?   [#permalink] 17 Dec 2018, 10:57
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