Author 
Message 
TAGS:

Hide Tags

Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1343
Location: Malaysia

What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
07 Feb 2017, 17:57
Question Stats:
62% (01:46) correct 38% (01:41) wrong based on 181 sessions
HideShow timer Statistics
What is the value of prime number \(p\)? (1) \(p+8\) is prime. (2) \(p−8\) is prime.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
"Be challenged at EVERY MOMENT."“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”"Each stage of the journey is crucial to attaining new heights of knowledge."Rules for posting in verbal forum  Please DO NOT post short answer in your post! Advanced Search : https://gmatclub.com/forum/advancedsearch/




Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4485

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
08 Feb 2017, 10:59
ziyuenlau wrote: What is the value of prime number \(p\)?
(1) \(p+8\) is prime. (2) \(p−8\) is prime. Dear ziyuenlau, I'm happy to respond. This is a devilishly clever and difficult problem. Let's jump to the case in which we are considering both statements together. Here are a few number sense rules. 1) In a set of three consecutive integers, one is always divisible by 3. 2) In a set of three evenly spaced integers, one is always divisible by 3. (These rules generalize from 3 to n.) The numbers (p  8), p, and (p + 8) are three evenly spaced numbers, so it absolutely must be true that one of them is divisible by 3. The only way all three of them can be prime is if the one that is divisible by 3 is 3 itself! Thus: p  8 = 3 p = 11 p + 8 = 19 If p takes the value p = 11, this is the only way that all three numbers can be prime simultaneously. Thus, the value of p is unique determined. Combined the statements are sufficient. OA = (C). Veritas wrote a truly beautiful problem here! Does all this make sense? Mike
_________________
Mike McGarry Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)




Math Expert
Joined: 02 Sep 2009
Posts: 64101

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
08 Feb 2017, 11:22
mikemcgarry wrote: ziyuenlau wrote: What is the value of prime number \(p\)?
(1) \(p+8\) is prime. (2) \(p−8\) is prime. Dear ziyuenlau, I'm happy to respond. This is a devilishly clever and difficult problem. Let's jump to the case in which we are considering both statements together. Here are a few number sense rules. 1) In a set of three consecutive integers, one is always divisible by 3. 2) In a set of three evenly spaced integers, one is always divisible by 3. (These rules generalize from 3 to n.) The numbers (p  8), p, and (p + 8) are three evenly spaced numbers, so it absolutely must be true that one of them is divisible by 3. The only way all three of them can be prime is if the one that is divisible by 3 is 3 itself! Thus: p  8 = 3 p = 11 p + 8 = 19 If p takes the value p = 11, this is the only way that all three numbers can be prime simultaneously. Thus, the value of p is unique determined. Combined the statements are sufficient. OA = (C). Veritas wrote a truly beautiful problem here! Does all this make sense? Mike This question might be inspired by the following two questions: https://gmatclub.com/forum/isthetwod ... 03359.htmlhttps://gmatclub.com/forum/istheposit ... 76553.html
_________________



Veritas Prep GMAT Instructor
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 95
Location: United States (DC)
GRE 1: Q170 V170 GRE 2: Q170 V170
GPA: 3.11
WE: Education (Education)

What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
08 Feb 2017, 16:43
Thank for the kind words, Mike! (This is one of mine.)
Bunuel, those weren't the inspiration, at least on a conscious level, but it's great to see the convergent evolution; they are certainly excellent and similar questions.
I'm very surprised this is currently listed as sub600 level. I debated how difficult to go with this one; I could have done something like p  8, p, p + 20 and gotten the same answer since p + 20 = p + 8 + 12 will be the same remainder w.r.t. 3 as p + 8.



Senior Manager
Joined: 21 Aug 2016
Posts: 253
Location: India
GPA: 3.9
WE: Information Technology (Computer Software)

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
09 Feb 2017, 09:07
mikemcgarry wrote: ziyuenlau wrote: What is the value of prime number \(p\)?
(1) \(p+8\) is prime. (2) \(p−8\) is prime. Dear ziyuenlau, I'm happy to respond. This is a devilishly clever and difficult problem. Let's jump to the case in which we are considering both statements together. Here are a few number sense rules. 1) In a set of three consecutive integers, one is always divisible by 3. 2) In a set of three evenly spaced integers, one is always divisible by 3. (These rules generalize from 3 to n.) The numbers (p  8), p, and (p + 8) are three evenly spaced numbers, so it absolutely must be true that one of them is divisible by 3. The only way all three of them can be prime is if the one that is divisible by 3 is 3 itself! Thus: p  8 = 3 p = 11 p + 8 = 19 If p takes the value p = 11, this is the only way that all three numbers can be prime simultaneously. Thus, the value of p is unique determined. Combined the statements are sufficient. OA = (C). Veritas wrote a truly beautiful problem here! Does all this make sense? Mike Hi mikemcgarry, I did not understand the rule well. When you say the numbers (p  8), p, and (p + 8) are three evenly spaced numbers, so it absolutely must be true that one of them is divisible by 3 even spaced means they are at the equal distance from preceding and succeeding term, or distance should be even as well? For instance 31 34 37 3 3 but none of them is divisible by 3; I think I misunderstood the concept. Please also explain how this rule is generalized from 3 to n. sorry for the silly question.



Veritas Prep GMAT Instructor
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 95
Location: United States (DC)
GRE 1: Q170 V170 GRE 2: Q170 V170
GPA: 3.11
WE: Education (Education)

What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
09 Feb 2017, 09:15
Good catch, AR15J! To clarify, the rule in question only applies to an evenly spaced set where the spacing isn't itself a multiple of 3.
More broadly, the general rule is this: Any list of N evenly spaced integers will contain a multiple of N as long as the spacing has no factors besides 1 in common with N. (In fact, it will contain exactly one such multiple.)
Try it!



Senior Manager
Joined: 21 Aug 2016
Posts: 253
Location: India
GPA: 3.9
WE: Information Technology (Computer Software)

What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
09 Feb 2017, 11:14
AnthonyRitz wrote: Good catch, AR15J! To clarify, the rule in question only applies to an evenly spaced set where the spacing isn't itself a multiple of 3.
More broadly, the general rule is this: Any list of N evenly spaced integers will contain a multiple of N as long as the spacing has no factors besides 1 in common with N. (In fact, it will contain exactly one such multiple.)
Try it! Thanks AnthonyRitz for clearing the doubt In GMAT also, evenly spaced means equal distance, below set is also evenly spaced?Please confirm 31,34,37



Veritas Prep GMAT Instructor
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 95
Location: United States (DC)
GRE 1: Q170 V170 GRE 2: Q170 V170
GPA: 3.11
WE: Education (Education)

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
09 Feb 2017, 13:05
AR15J wrote: AnthonyRitz wrote: Good catch, AR15J! To clarify, the rule in question only applies to an evenly spaced set where the spacing isn't itself a multiple of 3.
More broadly, the general rule is this: Any list of N evenly spaced integers will contain a multiple of N as long as the spacing has no factors besides 1 in common with N. (In fact, it will contain exactly one such multiple.)
Try it! Thanks AnthonyRitz for clearing the doubt In GMAT also, evenly spaced means equal distance, below set is also evenly spaced?Please confirm 31,34,37 AR15J, Yes, that is correct. 31, 34, 37 is evenly spaced. (But it will never contain a multiple of 3 no matter how long you continue, since the spacing is 3 and since at least one element is not a multiple of 3.) (But if, for instance, you include at least one more element in either direction, you will be guaranteed a multiple of 4.)



SVP
Joined: 24 Nov 2016
Posts: 1549
Location: United States

What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
15 Jun 2019, 06:01
AnthonyRitz wrote: Any list of N evenly spaced integers will contain a multiple of N as long as the spacing has no factors besides 1 in common with N. (In fact, it will contain exactly one such multiple.) Hello Bunuel could you verify this information, because the example below challenges what you AnthonyRitz just said. For N: 4, Set: {2,4,6,8}, Spacing/Difference: 2; We have a spacing (2) that is a factor of 4 and there is at least one multiple of 4 in the set. I think the rule is that for any set/list of N evenly spaced integers there is at least one multiple of N: (1) if any term is a multiple of N; or, (2) if the common difference (spacing) doesn't share any factors with N but 1.



GMAT Tutor
Joined: 24 Jun 2008
Posts: 2064

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
15 Jun 2019, 06:31
exc4libur wrote: AnthonyRitz wrote: Any list of N evenly spaced integers will contain a multiple of N as long as the spacing has no factors besides 1 in common with N. (In fact, it will contain exactly one such multiple.) AnthonyRitz wrote: Hello, could you verify the information below, because the example below challenges what you just said. For N: 4, Set: {2,4,6,8}, Spacing/Difference: 2; We have a spacing (2) that is a factor of 4 and there is at least one multiple of 4 in the set. What Anthony wrote is correct (though you wouldn't need to know it for the GMAT)  if you create an equally spaced list of N numbers that are d apart, then if the GCD of d and N is 1, the list will contain one multiple of N. The GCD of N and d in your list is not 1, so it's not a counterexample to that. I think you're trying to establish whether the logical converse of what Anthony wrote is also true, but in math, if you have a true statement, sometimes the converse is also true, and sometimes it's not true in general. And here, the converse is not true in general: when the GCD of N and d is not 1, then we need some more info to say how many multiples of N we'll have in our list. We can't take the converse (reverse) of the true fact above to conclude we should have zero multiples of N  that's only sometimes true. None of these facts are worth memorizing for the GMAT, though.
_________________
GMAT Tutor in Montreal
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com



SVP
Joined: 24 Nov 2016
Posts: 1549
Location: United States

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
15 Jun 2019, 07:12
IanStewart wrote: What Anthony wrote is correct (though you wouldn't need to know it for the GMAT)  if you create an equally spaced list of N numbers that are d apart, then if the GCD of d and N is 1, the list will contain one multiple of N.
The GCD of N and d in your list is not 1, so it's not a counterexample to that. I think you're trying to establish whether the logical converse of what Anthony wrote is also true, but in math, if you have a true statement, sometimes the converse is also true, and sometimes it's not true in general. And here, the converse is not true in general: when the GCD of N and d is not 1, then we need some more info to say how many multiples of N we'll have in our list. We can't take the converse (reverse) of the true fact above to conclude we should have zero multiples of N  that's only sometimes true.
None of these facts are worth memorizing for the GMAT, though. Hi Ian, Indeed, I was trying to establish the converse. Now I understand. Thks



Intern
Joined: 17 Aug 2019
Posts: 1

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
25 Aug 2019, 11:07
Hello, this is my first time posting a question here. I really need to understand why 5 is not a solution here, because in that case, it will satisfy the conditions of (p8) = 3, and (p+8) = 13, or is it that negative numbers are not considered as prime numbers?



Intern
Joined: 24 Jun 2017
Posts: 30

Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
Show Tags
26 Aug 2019, 22:43
Msfollie wrote: Hello, this is my first time posting a question here. I really need to understand why 5 is not a solution here, because in that case, it will satisfy the conditions of (p8) = 3, and (p+8) = 13, or is it that negative numbers are not considered as prime numbers? hi , prime no.'s are only "positive"by definition . so 3 is not a prime no . hope it helps




Re: What is the value of prime number p ? (1) p+8 is prime.
[#permalink]
26 Aug 2019, 22:43




