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

It is currently 18 Aug 2018, 16:56

Close

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
Your Progress

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.

Close

Request Expert Reply

Confirm Cancel

S97-10

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

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47983
S97-10  [#permalink]

Show Tags

New post 16 Sep 2014, 01:51
1
2
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

41% (01:26) correct 59% (01:28) wrong based on 93 sessions

HideShow timer Statistics

If \(n\) is a positive integer greater than 1, then \(p(n)\) represents the product of all the prime numbers less than or equal to \(n\). The second smallest prime factor of \(p(12) + 11\) is

A. 2
B. 11
C. 13
D. 17
E. 211

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47983
Re S97-10  [#permalink]

Show Tags

New post 16 Sep 2014, 01:51
Official Solution:

If \(n\) is a positive integer greater than 1, then \(p(n)\) represents the product of all the prime numbers less than or equal to \(n\). The second smallest prime factor of \(p(12) + 11\) is

A. 2
B. 11
C. 13
D. 17
E. 211


The quantity \(p(12)\) equals the product of all the primes less than or equal to 12. Thus, the number we are looking for is this:
\(2 \times 3 \times 5 \times 7 \times 11 + 11\)

\(= 11 \times (2 \times 3 \times 5 \times 7+1)\) [factor out the 11]

\(= 11 \times (210 + 1)\) [do the arithmetic]
\(= 11 \times 211\)
\(= \text{some large number}\)

We want to keep this number factored, and in fact we need to find its prime factorization. So, we ask: is 211 prime? Well, 211 cannot be divided evenly by 2, 3, 5, or 7, because 211 equals a multiple of all those numbers (210), plus 1. The "plus 1" means that 211 won't be divisible by any of the same factors as 210 (except for 1). Is 211 divisible by 11 or by 13? No, as we can check quickly by long division.

And we only need to check possible prime factors up to the square root of 211 (which is between 14, the square root of 196, and 15, the square root of 225). If there is a pair of factors besides 211 and 1, one of the factors in the pair must be lower than the square root of 211 (and the other factor in the pair would be larger). Since there are no prime factors of 211 less than 14, we know that 211 is prime, and the second smallest prime factor of \(p(12) + 11 = 11 \times 211\) is 211.


Answer: E
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
Joined: 15 Jul 2014
Posts: 10
Re: S97-10  [#permalink]

Show Tags

New post 24 Nov 2014, 19:20
Can someone please explain what does the question mean by second smallest prime factor and why 11 is not the right answer?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47983
Re: S97-10  [#permalink]

Show Tags

New post 25 Nov 2014, 05:02
1
kritiu wrote:
Can someone please explain what does the question mean by second smallest prime factor and why 11 is not the right answer?


Because 11 is the smallest prime. The second smallest prime is 211.

If n is a positive integer greater than 1, then p(n) represents the product of all the prime numbers less than or equal to n. The second smallest prime factor of p(12)+11 is

A. 2
B. 11
C. 13
D. 17
E. 211

\(p(12)+11=2*3*5*7*11+11=11(2*3*5*7+1)=11*211\). Both 11 and 211 are primes: 11 is the smallest prime of p(12)+11 and 211 is the second smallest prime of p(12)+11.

Answer: E.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Intern
Intern
avatar
Joined: 15 Jul 2014
Posts: 10
Re: S97-10  [#permalink]

Show Tags

New post 25 Nov 2014, 11:13
thanks Banuell. That helps!
Intern
Intern
User avatar
Status: Business Analyst
Joined: 17 Jul 2016
Posts: 10
Location: India
Re: S97-10  [#permalink]

Show Tags

New post 06 Feb 2017, 03:27
Hi

Why are we not considering 1 as one of the factors, then surely 11 would be the 2nd smallest?
Please correct me if i'm wrong.

Thanks
_________________

Regards,
Lumin

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47983
Re: S97-10  [#permalink]

Show Tags

New post 06 Feb 2017, 07:37
Senior SC Moderator
User avatar
V
Joined: 14 Nov 2016
Posts: 1322
Location: Malaysia
GMAT ToolKit User Premium Member CAT Tests
Re: S97-10  [#permalink]

Show Tags

New post 13 Feb 2017, 20:23
Bunuel wrote:
Official Solution:

If \(n\) is a positive integer greater than 1, then \(p(n)\) represents the product of all the prime numbers less than or equal to \(n\). The second smallest prime factor of \(p(12) + 11\) is

A. 2
B. 11
C. 13
D. 17
E. 211


The quantity \(p(12)\) equals the product of all the primes less than or equal to 12. Thus, the number we are looking for is this:
\(2 \times 3 \times 5 \times 7 \times 11 + 11\)

\(= 11 \times (2 \times 3 \times 5 \times 7+1)\) [factor out the 11]

\(= 11 \times (210 + 1)\) [do the arithmetic]
\(= 11 \times 211\)
\(= \text{some large number}\)

We want to keep this number factored, and in fact we need to find its prime factorization. So, we ask: is 211 prime? Well, 211 cannot be divided evenly by 2, 3, 5, or 7, because 211 equals a multiple of all those numbers (210), plus 1. The "plus 1" means that 211 won't be divisible by any of the same factors as 210 (except for 1). Is 211 divisible by 11 or by 13? No, as we can check quickly by long division.

And we only need to check possible prime factors up to the square root of 211 (which is between 14, the square root of 196, and 15, the square root of 225).If there is a pair of factors besides 211 and 1, one of the factors in the pair must be lower than the square root of 211 (and the other factor in the pair would be larger). Since there are no prime factors of 211 less than 14, we know that 211 is prime, and the second smallest prime factor of \(p(12) + 11 = 11 \times 211\) is 211.


Answer: E


Dear Bunuel, Do you have any example to illustrated the yellow highlight because I have no idea about what the statement is trying to convey.
_________________

"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/advanced-search/

CEO
CEO
User avatar
D
Joined: 12 Sep 2015
Posts: 2706
Location: Canada
Re: S97-10  [#permalink]

Show Tags

New post 02 May 2018, 09:23
Top Contributor
Bunuel wrote:
If \(n\) is a positive integer greater than 1, then \(p(n)\) represents the product of all the prime numbers less than or equal to \(n\). The second smallest prime factor of \(p(12) + 11\) is

A. 2
B. 11
C. 13
D. 17
E. 211


By the definition, p(12) = (11)(7)(5)(3)(2)

So, p(12) + 11 = (11)(7)(5)(3)(2) + 11
= 11[(7)(5)(3)(2) + 1] I factored out the 11
= 11[210 + 1]
= 11(211)

NOTE: 11 is prime and 211 is prime

The second smallest prime factor of p(12)+11 is . . .
We know that p(12)+11 has only 2 prime factors: 11 and 211
So, the smallest is 11, and the second smallest is 211

Answer: E

Cheers,
Brent
_________________

Brent Hanneson – Founder of gmatprepnow.com

Image

Re: S97-10 &nbs [#permalink] 02 May 2018, 09:23
Display posts from previous: Sort by

S97-10

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

Moderators: chetan2u, Bunuel

Events & Promotions

PREV
NEXT


cron

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.