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

 It is currently 20 Oct 2019, 07:45

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

The greatest common factor of 16 and the positive integer n

Author Message
TAGS:

Hide Tags

Manager
Joined: 28 Aug 2010
Posts: 151
The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

22 Feb 2011, 18:01
7
49
00:00

Difficulty:

75% (hard)

Question Stats:

59% (02:21) correct 41% (02:23) wrong based on 838 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
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9704
Location: Pune, India
Re: The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

22 Feb 2011, 19:38
12
3
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)

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

Director
Status: -=Given to Fly=-
Joined: 04 Jan 2011
Posts: 788
Location: India
Schools: Haas '18, Kelley '18
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
5
2
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'
_________________

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.
General Discussion
Manager
Joined: 28 Aug 2010
Posts: 151
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
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9704
Location: Pune, India
Re: The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

22 Feb 2011, 19:59
1
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

Manager
Joined: 28 Aug 2010
Posts: 151
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.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9704
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

Math Expert
Joined: 02 Sep 2009
Posts: 58434

Show Tags

25 Feb 2011, 08:32
4
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).

_________________
Manager
Joined: 17 Feb 2011
Posts: 143
Concentration: Real Estate, Finance
Schools: MIT (Sloan) - Class of 2014
GMAT 1: 760 Q50 V44

Show Tags

25 Feb 2011, 09:34
3
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: 22
Location: India
Concentration: Strategy, Sustainability
GMAT 1: 700 Q50 V34

Show Tags

04 Nov 2012, 06:43
1
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?
Senior Manager
Joined: 23 Mar 2011
Posts: 398
Location: India
GPA: 2.5
WE: Operations (Hospitality and Tourism)
Re: The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

20 Apr 2013, 12:29
1
mun23 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

Need easy explanation to solve it quickly

Hi, let me try to explain in simpler way:

GCF = 4 = 2^2
16 = 2^4
that means prime box of n = 2^2 , ? ? ?

GCF = 3
45 = 3^2*5
that means prime box of n = 3^1, ? ? ?

Overall prime box of n = 2^2, 3^1, ???

Now, 210 = 3*7*5*2
from above we know the prime factor and powers of n (not complete ??)
therefore GCF = 3^1 * 2^1 * 7^1 (not 5 - we have seen above, but at least a 7 is possible)
Thus at least a GCF of 42 is possible here

Hope this helps
_________________
"When the going gets tough, the tough gets going!"

Bring ON SOME KUDOS MATES+++

-----------------------------

My GMAT journey begins: http://gmatclub.com/forum/my-gmat-journey-begins-122251.html

Manager
Joined: 09 Apr 2013
Posts: 191
Location: United States
Concentration: Finance, Economics
GMAT 1: 710 Q44 V44
GMAT 2: 740 Q48 V44
GPA: 3.1
WE: Sales (Mutual Funds and Brokerage)
Re: The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

20 Apr 2013, 21:48
1
Simple explanation?

First, do a factor tree for each number.

You'll see that 5 can't be a factor of N (otherwise it would have been the highest factor between N and 45).

The highest factor that could theoretically exist between N and 210 is therefore all of the factors of 210 besides those we've ruled out. 210 is factored to 2 * 3 * 5 * 7. we've ruled out 5, so 2 * 3 * 7 = 42. Answer is D.

If the question asked "the highest factor that we KNOW exists" rather than "COULD" exist, the answer would be six since 2 and 3 are both factors of N, as well as of 210.
Manager
Joined: 10 Mar 2014
Posts: 180

Show Tags

21 Apr 2014, 08:35
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).

Hi Bunnel,

I have resolved this till n is a multiple of 2^2*3=12

why we are not considering 5 but we are considering 7.

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 58434

Show Tags

21 Apr 2014, 08:57
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).

Hi Bunnel,

I have resolved this till n is a multiple of 2^2*3=12

why 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.
_________________
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1749
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
1
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

_________________
Kindly press "+1 Kudos" to appreciate
Director
Joined: 17 Dec 2012
Posts: 626
Location: India
The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

01 Aug 2014, 23:49
n has to be a multiple of (2*2)*3 = 12
A common factor between 210= (2*3*5*7) and multiple of 12 is 2*3=6
So the G.C.F of n and 210 has to be a multiple of 6
The two choices that are multiples of 6 are 30 and 42.
Bur n is not a multiple of 5 .So 30 can be ruled out and the answer is 42.
_________________
Srinivasan Vaidyaraman
Sravna Test Prep
http://www.sravnatestprep.com

Holistic and Systematic Approach
Manager
Joined: 23 Dec 2013
Posts: 138
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
1
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.
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

01 Oct 2018, 16:26
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

$$?\,\,\,:\,\,\,GCF\left( {n\,,2 \cdot 3 \cdot 5 \cdot 7} \right)\,\,\underline {{\rm{could}}\,\,{\rm{be}}}$$

$$n \ge 1\,\,\,{\mathop{\rm int}}$$

$$GCF\left( {{2^4},n} \right) = {2^2}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{ {n \over {{2^2}}} = {\mathop{\rm int}} \hfill \cr {n \over {{2^{\, \ge \,3}}}} \ne {\mathop{\rm int}} \hfill \cr} \right.$$

$$GCF\left( {{3^2} \cdot 5,n} \right) = 3\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{ {n \over 3} = {\mathop{\rm int}} \hfill \cr {n \over {{3^{\, \ge \,2}}}} \ne {\mathop{\rm int}} \,\,\,\,\,;\,\,\,{n \over 5} \ne {\mathop{\rm int}} \,\, \hfill \cr} \right.$$

$$? = {2^1} \cdot {3^1} \cdot {7^{0\,{\rm{or}}\,1}}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{alternatives}}\,!} \,\,\,\,42\,\,\,\,\,\,\left( D \right)$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8109
Location: United States (CA)
Re: The greatest common factor of 16 and the positive integer n  [#permalink]

Show Tags

03 Oct 2018, 17:43
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

If the greatest common factor (GCF) of 16 and n is 4, n could be 4, 12, 20, 28, 36, etc. In other words, n is an odd multiple of 4.

Since the GCF of 45 and n is 3, and since 45 and 4 have no common factor other than 1, n must be a multiple of 3 x 4 = 12. However, since n is an odd multiple of 4, n actually has to be an odd multiple of 12 also.

If n = 12, we see that GCF(45, 12) = 3 and GCF(12, 210) = 6. However, 6 is not one of the choices.

If n = 36, we see that GCF(45, 36) = 9, but GCF(45, n) is supposed to be 3. So n can’t be 36.

If n = 60, we see that GCF(45, 60) = 15, but GCF(45, n) is supposed to be 3. So n can’t be 60.

If n = 84, we see that GCF(45, 84) = 3 and GCF(84, 210) = 42. We see that 42 is one of the choices.

_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Non-Human User
Joined: 09 Sep 2013
Posts: 13316

Show Tags

23 Sep 2019, 06:43
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.
_________________
Re: greatest common factor   [#permalink] 23 Sep 2019, 06:43
Display posts from previous: Sort by