GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 22 Sep 2019, 03:23 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. ### Request Expert Reply # What is the value of prime number p ? (1) p+8 is prime.

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Senior SC Moderator V
Joined: 14 Nov 2016
Posts: 1346
Location: Malaysia
What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

2
18 00:00

Difficulty:   55% (hard)

Question Stats: 63% (01:46) correct 37% (01:39) wrong based on 171 sessions

### HideShow timer Statistics

What is the value of prime number $$p$$?

(1) $$p+8$$ is prime.
(2) $$p−8$$ is prime.

_________________
"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!

Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4475
Re: What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

5
6
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 Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 58095
Re: What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

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/is-the-two-d ... 03359.html
https://gmatclub.com/forum/is-the-posit ... 76553.html
_________________
Veritas Prep GMAT Instructor G
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 45
Location: United States (DC)
GMAT 1: 790 Q51 V51 GRE 1: Q800 V800 GPA: 3.11
WE: Education (Education)
What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

1
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 sub-600 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  G
Joined: 21 Aug 2016
Posts: 258
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

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 G
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 45
Location: United States (DC)
GMAT 1: 790 Q51 V51 GRE 1: Q800 V800 GPA: 3.11
WE: Education (Education)
What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

1
2
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  G
Joined: 21 Aug 2016
Posts: 258
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

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 G
Affiliations: Veritas Prep
Joined: 21 Dec 2014
Posts: 45
Location: United States (DC)
GMAT 1: 790 Q51 V51 GRE 1: Q800 V800 GPA: 3.11
WE: Education (Education)
Re: What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

1
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.)
Senior Manager  G
Joined: 24 Nov 2016
Posts: 454
Location: United States
What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

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 G
Joined: 24 Jun 2008
Posts: 1836
Re: What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

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 Toronto

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
Senior Manager  G
Joined: 24 Nov 2016
Posts: 454
Location: United States
Re: What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

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

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 (p-8) = -3, and (p+8) = 13, or is it that negative numbers are not considered as prime numbers?
Intern  B
Joined: 24 Jun 2017
Posts: 28
Re: What is the value of prime number p ? (1) p+8 is prime.  [#permalink]

### Show Tags

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 (p-8) = -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, 23:43
Display posts from previous: Sort by

# What is the value of prime number p ? (1) p+8 is prime.

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  