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Is the positive integer P a prime number?

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Is the positive integer P a prime number? [#permalink] New post   18 Aug 2014, 07:59
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Is the positive integer P a prime number?

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


Source: 4Gmat
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C
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Re: Is the positive integer P a prime number? [#permalink] New post   18 Aug 2014, 08:41
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Is the positive integer P a prime number?

(1) (P+2) and (P+4) are prime. If p = 1, then the answer is NO but if p = 3, then the answer is YES. Not sufficient.

(2) (P-2) and (P-4) are prime. If p = 9, then the answer is NO but if p = 7, then the answer is YES. Not sufficient.

(1)+(2) Since (P-4), (P-2), (P+2), and (P+4) are primes, then all of them must be odd numbers. This means that (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers. Since there are no 5 consecutive odd primes, then P cannot be a prime (out of 5 consecutive odd numbers one must be a multiple of 3 and another a multiple of 5. There are only one prime, which is a multiple of 3, 3 itself and there are only one prime, which is a multiple of 5, 5 itself but P can be neither 3 or 5 since this violate (P-4), (P-2), (P+2), and (P+4) being primes). Sufficient.

Answer: C.

Hope it's clear.
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Is the positive integer P a prime number? [#permalink] New post   18 Aug 2014, 08:38
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For Stat1
P can be equal to 3 and can be equal to 9. So not sufficient

For STAT2
P can be equal to 7 and can be equal to 9. So not sufficient

STAT1 and STAT2 together
p-4, p-2,p, p+2, p+4
If P is prime then P must be odd (as only even prime is 2 and p-4 will not be prime then)
if p is odd then p cannot be prime as
p-4,p-2,p,p+2,p+4 make a series of 5 consecutive odd numbers and NO pair of 5 consecutive odd numbers are prime

So, p is NOT a prime number.
So, Answer will be C

Hope it helps!
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Re: Is the positive integer P a prime number? [#permalink] New post   18 Aug 2014, 08:43
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nktdotgupta wrote:
For Stat1
P can be equal to 2 and can be equal to 9. So not sufficient

For STAT2
P can be equal to 7 and can be equal to 9. So not sufficient

STAT1 and STAT2 together
p-4, p-2,p, p+2, p+4
If P is prime then P must be odd (as only even prime is 2 and p-4 will not be prime then)
if p is odd then p cannot be prime as
p-4,p-2,p,p+2,p+4 make a series of 5 consecutive odd numbers and NO pair of 5 consecutive odd numbers are prime

So, p is NOT a prime number.
So, Answer will be C

Hope it helps!


One little correction. P cannot be 2, since in this case neither P+2 nor P+4 is a prime.
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Re: Is the positive integer P a prime number? [#permalink] New post   18 Aug 2014, 09:14
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Thanks Bunuel, Wanted to write 3 :) (edited)
Bunuel wrote:
nktdotgupta wrote:
For Stat1
P can be equal to 2 and can be equal to 9. So not sufficient

For STAT2
P can be equal to 7 and can be equal to 9. So not sufficient

STAT1 and STAT2 together
p-4, p-2,p, p+2, p+4
If P is prime then P must be odd (as only even prime is 2 and p-4 will not be prime then)
if p is odd then p cannot be prime as
p-4,p-2,p,p+2,p+4 make a series of 5 consecutive odd numbers and NO pair of 5 consecutive odd numbers are prime

So, p is NOT a prime number.
So, Answer will be C

Hope it helps!


One little correction. P cannot be 2, since in this case neither P+2 nor P+4 is a prime.
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Re: Is the positive integer P a prime number? [#permalink] New post   06 Mar 2020, 01:06
Bunuel wrote:
Is the positive integer P a prime number?

(1) (P+2) and (P+4) are prime. If p = 1, then the answer is NO but if p = 3, then the answer is YES. Not sufficient.

(2) (P-2) and (P-4) are prime. If p = 9, then the answer is NO but if p = 7, then the answer is YES. Not sufficient.

(1)+(2) Since (P-4), (P-2), (P+2), and (P+4) are primes, then all of them must be odd numbers. This means that (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers. Since there are no 5 consecutive odd primes, then P cannot be a prime (out of 5 consecutive odd numbers one must be a multiple of 3 and another a multiple of 5. There are only one prime, which is a multiple of 3, 3 itself and there are only one prime, which is a multiple of 5, 5 itself but P can be neither 3 or 5 since this violate (P-4), (P-2), (P+2), and (P+4) being primes). Sufficient.

Answer: C.

Hope it's clear.


Hi Bunuel

I understand that the series of (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers, cannot be all odd. So by combination, the answer will be definite NO.

But I couldn't clearly understand the reason behind the numbers 3 and 5 you mentioned. Where does it come from and why just those 2.

Please help.
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Re: Is the positive integer P a prime number? [#permalink] New post   06 Mar 2020, 01:28
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Anurag06 wrote:
Bunuel wrote:
Is the positive integer P a prime number?

(1) (P+2) and (P+4) are prime. If p = 1, then the answer is NO but if p = 3, then the answer is YES. Not sufficient.

(2) (P-2) and (P-4) are prime. If p = 9, then the answer is NO but if p = 7, then the answer is YES. Not sufficient.

(1)+(2) Since (P-4), (P-2), (P+2), and (P+4) are primes, then all of them must be odd numbers. This means that (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers. Since there are no 5 consecutive odd primes, then P cannot be a prime (out of 5 consecutive odd numbers one must be a multiple of 3 and another a multiple of 5. There are only one prime, which is a multiple of 3, 3 itself and there are only one prime, which is a multiple of 5, 5 itself but P can be neither 3 or 5 since this violate (P-4), (P-2), (P+2), and (P+4) being primes). Sufficient.

Answer: C.

Hope it's clear.


Hi Bunuel

I understand that the series of (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers, cannot be all odd. So by combination, the answer will be definite NO.

But I couldn't clearly understand the reason behind the numbers 3 and 5 you mentioned. Where does it come from and why just those 2.

Please help.


The highlighted part explains why 5 consecutive odd numbers cannot all be primes. Out of 5 consecutive odd numbers, one must be a multiple of 3 and another must be a multiple of 5. So, IF all those 5 odd numbers are primes then one of them must be 3 (a multiple of 3) and another must be 5 (a multiple of 5). But if two of the 5 numbers is 3 and 5, then the remaining numbers cannot all be primes. Thus, not all 5 numbers are primes.
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Re: Is the positive integer P a prime number? [#permalink] New post   06 Mar 2020, 02:08
alphonsa wrote:
Is the positive integer P a prime number?

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


Source: 4Gmat


for condition 1;
(P+2) and (P+4) are prime
P can be 1,3,15 insufficient
from 2
(P-2) and (P-4) are prime
P can be 7,15 ; insufficient
from 1 &2
15 is common ;
IMO C sufficient
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Re: Is the positive integer P a prime number? [#permalink] New post   06 Mar 2020, 05:03
Bunuel wrote:
Anurag06 wrote:
Bunuel wrote:
Is the positive integer P a prime number?

(1) (P+2) and (P+4) are prime. If p = 1, then the answer is NO but if p = 3, then the answer is YES. Not sufficient.

(2) (P-2) and (P-4) are prime. If p = 9, then the answer is NO but if p = 7, then the answer is YES. Not sufficient.

(1)+(2) Since (P-4), (P-2), (P+2), and (P+4) are primes, then all of them must be odd numbers. This means that (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers. Since there are no 5 consecutive odd primes, then P cannot be a prime (out of 5 consecutive odd numbers one must be a multiple of 3 and another a multiple of 5. There are only one prime, which is a multiple of 3, 3 itself and there are only one prime, which is a multiple of 5, 5 itself but P can be neither 3 or 5 since this violate (P-4), (P-2), (P+2), and (P+4) being primes). Sufficient.

Answer: C.

Hope it's clear.


Hi Bunuel

I understand that the series of (P-4), (P-2), P, (P+2), and (P+4) are 5 consecutive odd numbers, cannot be all odd. So by combination, the answer will be definite NO.

But I couldn't clearly understand the reason behind the numbers 3 and 5 you mentioned. Where does it come from and why just those 2.

Please help.


The highlighted part explains why 5 consecutive odd numbers cannot all be primes. Out of 5 consecutive odd numbers, one must be a multiple of 3 and another must be a multiple of 5. So, IF all those 5 odd numbers are primes then one of them must be 3 (a multiple of 3) and another must be 5 (a multiple of 5). But if two of the 5 numbers is 3 and 5, then the remaining numbers cannot all be primes. Thus, not all 5 numbers are primes.



Thank you Bunuel for answering. Could you please let me know if there is any such property that says out of 5 consecutive numbers, one must be a multiple of 3 and 5 definitely but no other prime integers such as 7.
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Re: Is the positive integer P a prime number? [#permalink]
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