The greatest common factor of 16 and the positive integer n

Expert replies

Previous Topic Next Topic

BTGmoderatorLU: Moderator; Posts: 2544; Joined: Sun Oct 15, 2017 1:50 pmFollowed by: 6 members

by BTGmoderatorLU » Mon Oct 01, 2018 2:29 pm

Timer

00:00

Answers

Stats

Difficulty—

Source: Manhattan Prep

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 OA is D.

Like this post

Source: Beat The GMAT — Problem Solving | Mon Oct 01, 2018 2:29 pm

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Oct 01, 2018 3:19 pm

BTGmoderatorLU wrote:Source: Manhattan Prep

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

\[?\,\,\,:\,\,\,GCF\left( {n\,,2 \cdot 3 \cdot 5 \cdot 7} \right)\,\,\underline {{\text{could}}\,\,{\text{be}}} \]
\[n \geqslant 1\,\,\,\operatorname{int} \]
$$GCF\left( {{2^4},n} \right) = {2^2}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
{n \over {{2^2}}} = {\mathop{\rm int}} \hfill \cr
{n \over {{2^{\, \ge \,3}}}} \ne {\mathop{\rm int}} \hfill \cr} \right.$$
$$GCF\left( {{3^2} \cdot 5,n} \right) = 3\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
{n \over 3} = {\mathop{\rm int}} \hfill \cr
{n \over {{3^{\, \ge \,2}}}} \ne {\mathop{\rm int}} \,\,\,\,\,;\,\,\,{n \over 5} \ne {\mathop{\rm int}} \,\, \hfill \cr} \right.$$
$$? = {2^1} \cdot {3^1} \cdot {7^{0\,{\rm{or}}\,1}}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{alternatives}}\,!} \,\,\,\,42\,\,\,\,\,\,\left( D \right)$$

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

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

Like this post

GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Oct 01, 2018 4:52 pm

BTGmoderatorLU wrote:Source: Manhattan Prep

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 OA is D.

Now, we can use ELIMINATION to find the correct answer.

Goal: Find GCF of 210 and n.
210 = (2)(3)(5)(7)

The greatest common factor of 16 and the positive integer n is 4
16 = (2)(2)(2)(2)
4 = (2)(2)
So, we know for certain that the PRIME FACTORIZATION of n has TWO 2's (and no more than TWO)
In other words, n = (2)(2)(?)(?)...
Since 210 = (2)(3)(5)(7), we can see that 210 and n both share ONE 2.
So, the GCF of 210 and n will be divisible by 2.
We can ELIMINATE A, since 3 is not divisible by 2.

The greatest common factor of n and 45 is 3
45 = (3)(3)(5)
3 = 3
So, we know for certain that the PRIME FACTORIZATION of n has ONE 3 (and no more than ONE)
In other words, n = (3)(?)(?)(?)...
Since 210 = (2)(3)(5)(7), we can see that 210 and n both share ONE 3.
So, the GCF of 210 and n will be divisible by 3.
We can ELIMINATE B and E, since they are not divisible by 3.

This leaves C (30) and D (42).

Finally, notice that, since the GCF of n and 45 is 3, we can be certain that n does NOT have a 5 in its prime factorization. Otherwise, the GCF of n and 45 would be 15.
This means that n is NOT divisible by 5, which also means the GCF of 210 and n will NOT be divisible by 5
So we can ELIMINATE C.

This leaves us with D, the correct answer.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

Like this post

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Oct 01, 2018 5:29 pm

BTGmoderatorLU wrote:Source: Manhattan Prep

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 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3.
Draw a VENN DIGRAM showing where the prime factors of n, 16 and 45 overlap:

The diagram implies the following:
Since n and 16 have only two 2's in common, the prime-factorization of n includes exactly two 2's.
Since n and 45 have only a 3 in common, the prime-factorization of n includes exactly one 3 and no 5's.
Thus:
n = 2*2*3*k, where k is not a multiple of 2, 3, or 5.

Since n is not a multiple of 5, the GCF of n and 210 cannot be a multiple of 5.
Eliminate C and E.
Since n = 2*2*3*k and 210 = 2*3*5*7, the GCF of n and 210 must be a multiple of 6, as indicated by the factors in red.
Eliminate A and B.

The correct answer is D.

Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

Like this post

GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Wed Oct 03, 2018 4:40 pm

BTGmoderatorLU wrote:Source: Manhattan Prep

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.

Answer: D

Scott Woodbury-Stewart
Founder and CEO
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

559 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review