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Is the two digit positive integer P a prime number?
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Updated on: 14 Sep 2014, 17:29
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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, 17:01.
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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Is the two digit positive integer P a prime number?
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21 Oct 2010, 17:52
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: {P2}{P1}{P}{P+1}{P+2}. Now, P2 and P+2 are not divisible by 3 as they are primes, P+1 and P1 are 3 apart from P2 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: {P1}{P}{P+1}. P1 and P+1 are 3 apart from P4 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?
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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 P2 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?
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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 & P2 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 & P4 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?
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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?
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Re: Is the two digit positive integer P a prime number?
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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 6n1 or 6n+1 from1 6n1+2 = 6n+1 and 6n12 = 6n3 = 3(2n1) also if we try 6n+1 again if p is prime then it is impossible for p+2 and p2 for both tp be prime thus p aint prime from 2 using the same way 6n14 = 6n5 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?
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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 6n1 or 6n+1 from1 6n1+2 = 6n+1 and 6n12 = 6n3 = 3(2n1) also if we try 6n+1 again if p is prime then it is impossible for p+2 and p2 for both tp be prime thus p aint prime from 2 using the same way 6n14 = 6n5 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?
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05 May 2018, 15:07
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 p2 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?
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11 Jul 2018, 23:37
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 P2 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. evenodd=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 P4 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. evenodd = 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?
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11 Jul 2018, 23:48
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 P2 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. evenodd=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 P4 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. evenodd = 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 twodigit prime, so odd, then P is odd not even: odd + 2 = odd + even = odd = P. Similarly, if P + 4 (or P  4) is a twodigit 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?
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11 Jul 2018, 23:59
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?
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12 Jul 2018, 00:30
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 6k1 (actually all primes except 2 & 3 are of that form) statement 1) p+2 & p2 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 6k1 then p2 is of the form 6k3 which cannot be a prime so p is not a prime of form 6k1 hence, p is not a prime. sufficient. statement 2) p+4 & p4 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 6k1 then p+4 is of the form 6k+3 which cannot be prime so p is not a prime of form 6k1 hence, p is not a prime. sufficient. D.
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