Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 28 Aug 2010
Posts: 222

The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
22 Feb 2011, 18:01
Question Stats:
58% (01:42) correct 42% (01:28) wrong based on 623 sessions
HideShow timer Statistics
The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210? A. 3 B. 14 C. 30 D. 42 E. 70
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Verbal:newtotheverbalforumpleasereadthisfirst77546.html Math: newtothemathforumpleasereadthisfirst77764.html Gmat: everythingyouneedtoprepareforthegmatrevised77983.html  Ajit




Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8282
Location: Pune, India

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
22 Feb 2011, 19:38
ajit257 wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
3
14
30
42
70
i am not so sure about the oa. GCF (n, 16) = 4 This means 4 is a factor of n but 8 and 16 are not. (If 8 were a factor of n too, the GCF would have been 8. Similarly for 16) GCF (n, 45) = 3 This means 3 is a factor of n but 9 and 5 are not. Same logic as above. 210 = 2*3*5*7 n has 4 and 3 as factors and it doesn't have 5 as a factor. so GCF of n and 210 could be 6 (if 7 is not a factor of n) or 42 (if 7 is a factor of n) Answer (D) Note: 3 is definitely not the GCF of n and 210 because they definitely have 3*2 in common. So GCF has to be at least 6.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!




Director
Status: =Given to Fly=
Joined: 04 Jan 2011
Posts: 816
Location: India
Concentration: Leadership, Strategy
GMAT 1: 650 Q44 V37 GMAT 2: 710 Q48 V40 GMAT 3: 750 Q51 V40
GPA: 3.5
WE: Education (Education)

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
Updated on: 22 Feb 2011, 19:51
EDIT.... My explanation was wrong :p [Correcting it] Corrected Version: Let's try the Prime Box Approach Prime Box is simply a collection of all prime factors of a given number! (1) Prime Box of 16 = 2, 2, 2, 2 (2) Prime Box of 45 = 3, 3, 5 (3) Prime Box of n = 2, 2, 3.... (4) Prime Box of 210 = 2, 5, 3, 7 From 3 and 4: The GCF of n and 210 must be a multiple of 6. So we can eliminate A, B and E! From 2 and 3: n is not a multiple of 5. If it were, the GCF of n and 45 would have been 15! So we can eliminate C The only remaining choice is 'D'
_________________
"Wherever you go, go with all your heart"  Confucius
Useful Threads
1. How to Review and Analyze your Mistakes (Post by BB at GMAT Club)
2. 4 Steps to Get the Most out out of your CATs (Manhattan GMAT Blog)
My Experience With GMAT
1. From 650 to 710 to 750  My Tryst With GMAT
2. Quest to do my Best  My GMAT Journey Log
Originally posted by AmrithS on 22 Feb 2011, 19:36.
Last edited by AmrithS on 22 Feb 2011, 19:51, edited 2 times in total.




Manager
Joined: 28 Aug 2010
Posts: 222

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
22 Feb 2011, 19:54
thanks Karishma ...my doubt is exactly the same thing but for 16 ..how come 42 is correct if n and 16 have a gcf of 4
_________________
Verbal:newtotheverbalforumpleasereadthisfirst77546.html Math: newtothemathforumpleasereadthisfirst77764.html Gmat: everythingyouneedtoprepareforthegmatrevised77983.html  Ajit



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8282
Location: Pune, India

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
22 Feb 2011, 19:59
ajit257 wrote: thanks Karishma ...my doubt is exactly the same thing but for 16 ..how come 42 is correct if n and 16 have a gcf of 4 That is because 210 has only one 2. Even though n has a 4 as factor, 210 does not. Therefore GCF of n and 210 does not have 4 as a factor. Does it make sense now?
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Manager
Joined: 28 Aug 2010
Posts: 222

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
22 Feb 2011, 20:01
so lets say if another choice was given as 6 then what could have been the ans or it would not be that close.
_________________
Verbal:newtotheverbalforumpleasereadthisfirst77546.html Math: newtothemathforumpleasereadthisfirst77764.html Gmat: everythingyouneedtoprepareforthegmatrevised77983.html  Ajit



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8282
Location: Pune, India

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
22 Feb 2011, 20:09
ajit257 wrote: so lets say if another choice was given as 6 then what could have been the ans or it would not be that close. Then there would have been two correct options: 6 and 42. Either can be the GCF of n and 210 depending on what exactly n is. GMAT never has 2 correct options and hence such a scenario is not possible. Only one of 6 and 42 would be in the answer choices.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Math Expert
Joined: 02 Sep 2009
Posts: 49271

rosgmat wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
a) 3 b) 14 c) 30 d) 42 e) 70 The greatest common factor of 2^4=16 and n is 4 > n is a multiple of 2^2=4 but not the higher powers of 2, for example 2^3=8 or 2^4=16, because if it were then the greatest common factor of 16 and n would be more than 4; The greatest common factor of 3^2*5=45 and n is 3 > n is a multiple of 3 but not the higher powers of 3 and not 5, because if it were then the greatest common factor of 3^2*5=45 and n would be more than 3; So, n is a multiple of 2^2*3=12, not a multiple of higher powers of 2 or 3, and not a multiple of 5 (so n=12x where x could be any positive integer but 2, 3, or 5). Now, as 210=2*3*5*7 then the greatest common factor of n and 210 could be 6 or 6*7=42 (if 7 is a factor of n). Answer: D.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 17 Feb 2011
Posts: 151
Concentration: Real Estate, Finance
Schools: MIT (Sloan)  Class of 2014

You can do the prime boxes.
Prime box of 16: 2, 2, 2, 2 Prime box of 45: 3, 3, 5
Prime box of 210: 2, 5, 3, 7
So, n has at least two 2's and one 3, but n hasn't got any 5. Now, checking alternatives: A) wrong, as n and 210 share at least one 2 and one 3. B) wrong again, no 3 in 14. C) wrong, as 30 has a 5 D) correct. 42 prime box is 2, 3, 7, so it meets all requirements. E) wrong, 70 prime box has 2, 7 and 5



Intern
Joined: 09 Jul 2012
Posts: 23
Location: India
Concentration: Strategy, Sustainability

Question on GCF
[#permalink]
Show Tags
04 Nov 2012, 06:43
The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?



Intern
Joined: 09 Jul 2012
Posts: 23
Location: India
Concentration: Strategy, Sustainability

Re: Question on GCF
[#permalink]
Show Tags
04 Nov 2012, 06:49
the Common GCF of 16 and n being 4, made me choose n to be 12. the Common GCF of n and 45 being 3, n= 12 seems to be a valid option here as well. Hence, the Common GCF of n and 210, i.e. 12 and 210 seems to be 6.



Intern
Joined: 07 Aug 2012
Posts: 21

Re: Question on GCF
[#permalink]
Show Tags
04 Nov 2012, 08:12
Some2609 wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210? Haha, I just clicked wrong in the poll, but imho here goes the correct way: 16 and n  GCF = 4 = 2 x 2 45 and n  GCF = 3 210 = 2 x 3 x 5 x 7 Eliminate prime factors that are not included in the given options and approve the ones that appear. Eliminate: 5 Approve: 2, 3, 7 2 x 3 x 7 = 42
_________________
Exhaust your body, proceed your mind, cultivate your soul.



Manager
Joined: 10 Mar 2014
Posts: 205

Bunuel wrote: rosgmat wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
a) 3 b) 14 c) 30 d) 42 e) 70 The greatest common factor of 2^4=16 and n is 4 > n is a multiple of 2^2=4 but not the higher powers of 2, for example 2^3=8 or 2^4=16, because if it were then the greatest common factor of 16 and n would be more than 4; The greatest common factor of 3^2*5=45 and n is 3 > n is a multiple of 3 but not the higher powers of 3 and not 5, because if it were then the greatest common factor of 3^2*5=45 and n would be more than 3; So, n is a multiple of 2^2*3=12, not a multiple of higher powers of 2 or 3, and not a multiple of 5 (so n=12x where x could be any positive integer but 2, 3, or 5). Now, as 210=2*3*5*7 then the greatest common factor of n and 210 could be 6 or 6*7=42 (if 7 is a factor of n). Answer: D. Hi Bunnel, I have resolved this till n is a multiple of 2^2*3=12why we are not considering 5 but we are considering 7. Thanks



Math Expert
Joined: 02 Sep 2009
Posts: 49271

PathFinder007 wrote: Bunuel wrote: rosgmat wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
a) 3 b) 14 c) 30 d) 42 e) 70 The greatest common factor of 2^4=16 and n is 4 > n is a multiple of 2^2=4 but not the higher powers of 2, for example 2^3=8 or 2^4=16, because if it were then the greatest common factor of 16 and n would be more than 4; The greatest common factor of 3^2*5=45 and n is 3 > n is a multiple of 3 but not the higher powers of 3 and not 5, because if it were then the greatest common factor of 3^2*5=45 and n would be more than 3; So, n is a multiple of 2^2*3=12, not a multiple of higher powers of 2 or 3, and not a multiple of 5 (so n=12x where x could be any positive integer but 2, 3, or 5). Now, as 210=2*3*5*7 then the greatest common factor of n and 210 could be 6 or 6*7=42 (if 7 is a factor of n). Answer: D. Hi Bunnel, I have resolved this till n is a multiple of 2^2*3=12why we are not considering 5 but we are considering 7. Thanks Notice that the GCF of 45 (a multiple of 5) and n is 3 (not a multiple of 5). This means that n itself cannot be a multiple of 5. As for 7: we know for sure that 2 and 3 are factors of n and 5 is not a factor of n. We know nothing about its other primes, so any prime greater than 5 theoretically can be a factor of n.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 23 Apr 2014
Posts: 11
Location: India

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
03 Jun 2014, 06:56
1st pair (n and 16) for whom the GCF is 4
GCF=4=2^2 16=2^4 Since GCF contains the lowest powers of all the common prime factors it can be deducted that n must contain 2^2
2nd pair (n and 45)for whom the GCF is 3 GCF=3=3^1 45=3^(2 ) x 5^1 Since GCF contains the lowest powers of all the common prime factors it can be deducted that n must contain 3^1 and must not contain 5^1
3rd Pair (n and 210) 210=2 x 3x 5 x 7 n=must contain 2^2,must contain 3^1,may contain 7,must not contain 5 Therefore n could be either=(2^2 x 3^1=12)or (2^2 x 3^1 x 7^1=84)
if n=12 then GCF of 12 and 210 is 2 x 3=6 if n=84 then GCF of 84 and 210 is 2 x 3 x 7=42
42 is the correct answer



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1834
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
04 Jun 2014, 00:43
16........ n .......................... n ....... 45 GCF = 4 ................................. GCF = 3 So n may be either 4 * 3 = 12 or a multiple of 12 to satisfy the GCF values given. 210 = 7 * 2 * 5 * 3 2, 3 & 5 are associated with 16 & 45. Including them in calculations will disturb the already provided GCF values Only 7 stands out. So 12 * 7 = 84 GCF of 84 & 210 = 42 Answer = D
_________________
Kindly press "+1 Kudos" to appreciate



Intern
Joined: 02 Mar 2015
Posts: 31

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
04 Aug 2015, 04:16
ajit257 wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
A. 3 B. 14 C. 30 D. 42 E. 70 GCF (n, 16) = 4 GCF (n, 45) = 3 so n only contains 4 and 3 no 5 210 = 2*3*5*7 lowest GCF is 6 possible GCF in answer choice is 42 Gib kudos



Current Student
Joined: 12 Aug 2015
Posts: 2651

The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
29 Jan 2017, 04:20
This is an excellent Question. Here is what i did in this question >
GCD(N,16)=4 Hence N=2^2*k where 2 is not the prime factor of k. GCD(N,45)=3 N=3*m where 3 and 5 are not prime factors of m.
Combing the two equations above > N=2^2*3*x where 2,3,5 are not prime factors of x.
Now 210=2*3*5*7 Hence GCD (N,210)=2*3*common factors among x and 7 So GCD must be a multiple of 6. But wait a second. There are two options which are multiples of 6. Notice for GCD to be 30 => x must have 5 as its prime. But x cannot have 5 as its prime. Hence 30 is out too.
So the only viable choice is 42. As it turns out if x=7 => GCD(N,210)=42
Hence D.
_________________
MBA Financing: INDIAN PUBLIC BANKS vs PRODIGY FINANCE! Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs!STONECOLD's BRUTAL Mock Tests for GMATQuant(700+)AVERAGE GRE Scores At The Top Business Schools!



Manager
Joined: 23 Dec 2013
Posts: 182
Location: United States (CA)
GMAT 1: 710 Q45 V41 GMAT 2: 760 Q49 V44
GPA: 3.76

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
04 Jun 2017, 19:07
ajit257 wrote: The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?
A. 3 B. 14 C. 30 D. 42 E. 70 This problem works best if you break it down into pieces with Venn diagrams. 16 and n have a GCF of 4 = 2^2. That means n has at least two 2's as factors. 45 and n share one factor: 3. That means n has at least one 3 and two 2's (3*2^2). The next step is to break down 210 into 2*3*7*5. 5 is not a possible factor of n because it would've been a GCF with 45 and n. But 7 could be a common divisor because it's not expressly forbidden anywhere. So the GCF in this case is 2*3*7 = 42.



NonHuman User
Joined: 09 Sep 2013
Posts: 8105

Re: The greatest common factor of 16 and the positive integer n
[#permalink]
Show Tags
25 Jun 2018, 03:57
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: The greatest common factor of 16 and the positive integer n &nbs
[#permalink]
25 Jun 2018, 03:57






