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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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New post Updated on: 14 Sep 2014, 16:29
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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.

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Originally posted by gurpreetsingh on 21 Oct 2010, 16:01.
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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Is the two digit positive integer P a prime number?  [#permalink]

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New post 21 Oct 2010, 16:52
5
11
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 cannot be a prime. Sufficient.

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

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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]

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New post 22 Oct 2010, 12:09
1
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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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]

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

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New post 14 Jan 2017, 07:51
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

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

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New post 14 Jan 2017, 07:53
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

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

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New post 05 May 2018, 14:07
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.


Hi,
Fastest way possible,,,, try urself
write down all prime numbers till 50( 11,13,17,19,23,29,31,37,43,47,......)
Statement 1 : p+2 and p-2 are prime means integer p is halfway between two primes that are 4 distances apart on number line....(e.g. 13 and 17 , 43 and 47 so on...) .. in any case, p is not a prime. Hence A is sufficient.
Statement 2 : Apply similar logic. (possible pairs are 11 and 19, 29 and 37 so on.....) Hence B is Sufficient..
Therefore, Ans is D
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 22:37
1
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.


The question can be solved in less than 1 minute.

Statement 1.
Since P+2 and P-2 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even+odd = odd. even-odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Statement 2.
Since P-4 and P+4 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even-odd = odd. even+odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Hence option D is the answer.
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 22:48
1
houston1980 wrote:
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.


The question can be solved in less than 1 minute.

Statement 1.
Since P+2 and P-2 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even+odd = odd. even-odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Statement 2.
Since P-4 and P+4 are primes, and P is a two digit positive integer , then P must be even.
This is because all two digit positive integer prime numbers are odd.
even-odd = odd. even+odd=odd. Therefore P must be two digit even number and therefore P is not prime. 2 is the only even prime number.

Hence option D is the answer.


That's not correct.

If P + 2 (or P - 2) is a two-digit prime, so odd, then P is odd not even: odd + 2 = odd + even = odd = P.

Similarly, if P + 4 (or P - 4) is a two-digit prime, so odd, then P is odd not even: odd + 4 = odd + even = odd = P.

The question is not that easy. You can check correct solutions in the discussion above.

Hope it helps.
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Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 22:59
I agree with you Bunuel. You are correct, I was wrong. My solutions was incorrect. Thanks Bunuel!!!
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Re: Is the two digit positive integer P a prime number?  [#permalink]

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New post 11 Jul 2018, 23:30
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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.


all 2 digit primes are either of the form 6k+1 or 6k-1 (actually all primes except 2 & 3 are of that form)

statement 1) p+2 & p-2 are prime
if p is prime of the form 6k+1
then p+2 is of form 6k+3 which can not be a prime so p is not a prime of form 6k+1
if p is prime of the form 6k-1
then p-2 is of the form 6k-3 which cannot be a prime so p is not a prime of form 6k-1
hence, p is not a prime. sufficient.

statement 2) p+4 & p-4 are prime
if p is prime of the form 6k+1
then p+4 is of the form 6k+5 which cannot be prime so p is not a prime of form 6k+1
if p is prime of the form 6k-1
then p+4 is of the form 6k+3 which cannot be prime so p is not a prime of form 6k-1
hence, p is not a prime. sufficient.

D.
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Re: Is the two digit positive integer P a prime number? &nbs [#permalink] 11 Jul 2018, 23:30
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