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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]

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21 Oct 2010, 17:01
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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, 17:29, 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]

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21 Oct 2010, 17:52
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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.

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Re: Is the two digit positive integer P a prime number? [#permalink]

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21 Oct 2010, 18:49
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]

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22 Oct 2010, 13:09
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]

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22 Oct 2010, 22:53
Ohh yes... I missed many pairs...
gurpreetsingh , Thanks for the correction
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Re: Is the two digit positive integer P a prime number? [#permalink]

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14 Sep 2014, 09:30
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Re: Is the two digit positive integer P a prime number? [#permalink]

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14 Jan 2017, 08:00
Hi chetan2u

Any quickies to solve this one ?
Since i remember the primes upto 100 i was able to arrive at an answer,but only after a good 3 minutes.

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Re: Is the two digit positive integer P a prime number? [#permalink]

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14 Jan 2017, 08:51
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.

any prime >=5 is in the form 6n-1 or 6n+1

from1

6n-1+2 = 6n+1 and 6n-1-2 = 6n-3 = 3(2n-1) also if we try 6n+1 again if p is prime then it is impossible for p+2 and p-2 for both tp be prime thus p aint prime

from 2

using the same way

6n-1-4 = 6n-5 and 6n+3 = 3(2n+1) ... again p is impossible to be prime

D
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Re: Is the two digit positive integer P a prime number? [#permalink]

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14 Jan 2017, 08:53
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.

any prime >=5 is in the form 6n-1 or 6n+1

from1

6n-1+2 = 6n+1 and 6n-1-2 = 6n-3 = 3(2n-1) also if we try 6n+1 again if p is prime then it is impossible for p+2 and p-2 for both tp be prime thus p aint prime

from 2

using the same way

6n-1-4 = 6n-5 and 6n+3 = 3(2n+1) ... again p is impossible to be prime

D
Re: Is the two digit positive integer P a prime number?   [#permalink] 14 Jan 2017, 08:53
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