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Is the two digit positive integer P a prime number?

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Is the two digit positive integer P a prime number? [#permalink] New post 21 Oct 2010, 16:01
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Is the two digit positive integer P a prime number?

(1) P + 2 and P – 2 are prime.
(2) P – 4 and P + 4 are prime.
[Reveal] Spoiler: OA

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Last edited by Bunuel on 14 Sep 2014, 16:29, edited 1 time in total.
Edited the question and added the OA.
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Re: Is the two digit positive integer P a prime number? [#permalink] New post 21 Oct 2010, 16:52
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gurpreetsingh wrote:
Is the two digit positive integer P a prime number?

1) P + 2 and P – 2 are prime.
2) P – 4 and P + 4 are prime.

I will post the solution and the explanation later.


Good question, +1.

Note that P is the two digit integer .

(1) P + 2 and P – 2 are prime --> out of every 3 consecutive integers one must be divisible by 3 --> consider 5 consecutive integers: {P-2}{P-1}{P}{P+1}{P+2}. Now, P-2 and P+2 are not divisible by 3 as they are primes, P+1 and P-1 are 3 apart from P-2 and P+2 respectively and thus they are also not divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it can not be a prime. Sufficient.

(2) P – 4 and P + 4 are prime --> the same here: {P-1}{P}{P+1}. P-1 and P+1 are 3 apart from P-4 and P+4 respectively and thus can not be divisible by 3, which means that P must be divisible by 3 and as it's TWO digit integer it can not be a prime. Sufficient.

Answer: D.
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Re: Is the two digit positive integer P a prime number? [#permalink] New post 21 Oct 2010, 17:49
Yes Bunuel, Its a very good question. That's why I thought I should share it here.

The only catch was the existence of two digit number. Since you have posted the solution. I will post mine strategy.

All the numbers can be represented as 3k,3k+1, 3k+2

Statement 1: For P+2 and P-2 to be prime the number can not be of the form 3k+1 and 3k+2.
=> P = 3k = not prime if k is not equal to 1
Since two digit number is given => 3k is not a prime number.

Statement 2: using the same strategy. P is not a prime number.

Hence D.
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Re: Is the two digit positive integer P a prime number? [#permalink] New post 22 Oct 2010, 12:09
I solved this by different method,

1) P + 2 and P – 2 are prime.
--> if You can list out all two digit Prime Numbers, you will get P=15 ONLY for which P+2 & P-2 are primes.
since 15 is not prime, Sufficient

2) P – 4 and P + 4 are prime.
--> Again, if You can list out all two digit Prime Numbers, you will get P=15 ONLY for which P+4 & P-4 are primes.
since 15 is not prime, Sufficient

hence D.

though the answer is correct by this method, I feel Bunuel's Method was more proper !! :)
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Re: Is the two digit positive integer P a prime number? [#permalink] New post 22 Oct 2010, 21:53
Ohh yes... I missed many pairs... :o
gurpreetsingh , Thanks for the correction
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Re: Is the two digit positive integer P a prime number? [#permalink] New post 14 Sep 2014, 08:30
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Is the two digit positive integer P a prime number? [#permalink] New post 06 Aug 2015, 12:41
Is the two digit positive integer P a prime number?

(1) (P + 2) and (P - 2) are prime.

(2) (P - 4) and (P + 4) are prime.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
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Re: Is the two digit positive integer P a prime number? [#permalink] New post 06 Aug 2015, 12:56
Tornikea wrote:
Is the two digit positive integer P a prime number?

(1) (P + 2) and (P - 2) are prime.

(2) (P - 4) and (P + 4) are prime.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.


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Re: Is the two digit positive integer P a prime number?   [#permalink] 06 Aug 2015, 12:56
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