GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
close
It is currently 22 Jun 2021, 22:41
Register
GMAT Club Tests
Decision Tracker
My Rewards
New posts
New comers' posts

MBA Spotlight – Top 20 Virtual MBA Fair
June 14-30, 2021
Register Schedule
Super Wednesday:
INSEAD AdCom Live Q&A MIT Sloan AdCom Live Q&A Goizueta AdCom Live Q&A How to Pay for Your MBA Landing a Tech PM Job
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.

Go to My Error Log Learn more
Close

Request Expert Reply

Confirm Cancel
What is the greatest prime factor of 4^17 - 2^28?
26 posts Go to First Unread Post
Print view
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
SORT BY:
Date
Kudos
 1   2   
Tags:
Find Similar Topics 

Show Tags
Hide Tags

User avatar
student26
Intern
Intern
Joined: 25 Oct 2010
Posts: 21
WE 1: 3 yrs
What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  13 Nov 2010, 09:55
11
Kudos
117
Bookmarks
Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
00:00
A
B
C
D
E

Difficulty:

25% (medium)

Question Stats:

71% (01:06) correct 29% (01:17) wrong based on 2085 sessions

Hide Show timer Statistics

What is the greatest prime factor of \(4^{17} - 2^{28}\)?

A. 2
B. 3
C. 5
D. 7
E. 11
ShowHide Answer
Official Answer
Official Answer and Stats are available only to registered users. Register/Login.
Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
Most Helpful Expert Reply
User avatar
V
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 77369
Premium Member CAT Tests Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  13 Nov 2010, 10:03
19
Kudos
44
Bookmarks
Expert Reply
student26 wrote:
What is the greatest prime factor of 4^17 - 2^28?

A. 2
B. 3
C. 5
D. 7
E. 11


\(4^{17}-2^{28}=2^{34}-2^{28}=2^{28}(2^6-1)=2^{28}*63=2^{28}*3^2*7\) --> greatest prime factor is 7.

Answer: D.
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
General Discussion
User avatar
rockzom
Manager
Manager
Joined: 25 Oct 2010
Status:Planning to retake.
Affiliations: Alpha Psi Omega
Posts: 81
Concentration: General Management, Entrepreneurship
GMAT 1: 650 Q42 V37
GRE 1: Q630 V680
GPA: 3.16
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  13 Nov 2010, 11:40
1
Kudos
I have a hard time ignoring that 2^28 and trusting that breaking down 63 into its prime factors will give me the right answer.

Is there a hard and fast rule for why you can forget about 2^28 and trust that the 7 derived from 63 is correct?
User avatar
V
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 77369
Premium Member CAT Tests Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  13 Nov 2010, 11:47
1
Bookmarks
Expert Reply
rockzom wrote:
I have a hard time ignoring that 2^28 and trusting that breaking down 63 into its prime factors will give me the right answer.

Is there a hard and fast rule for why you can forget about 2^28 and trust that the 7 derived from 63 is correct?


\(4^{17}-2^{28}\) equals to \(2^{28}*3^2*7\), which means that the prime factors of this number are 2, 3, and 7, so the greatest prime factor is 7 (2^28=2*2*...*2, so this expression has only one prime: 2).
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
rockzom
Manager
Manager
Joined: 25 Oct 2010
Status:Planning to retake.
Affiliations: Alpha Psi Omega
Posts: 81
Concentration: General Management, Entrepreneurship
GMAT 1: 650 Q42 V37
GRE 1: Q630 V680
GPA: 3.16
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  13 Nov 2010, 12:05
I see. So essentially any time that you have an expression where all of the bases are prime, you can assume that the highest base would be the greatest prime factor?

For example, the expression \(7^7 * 13^3 * 17\) would have 17 as the greatest prime factor. Correct?
User avatar
V
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 77369
Premium Member CAT Tests Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  13 Nov 2010, 12:10
1
Kudos
Expert Reply
rockzom wrote:
I see. So essentially any time that you have an expression where all of the bases are prime, you can assume that the highest base would be the greatest prime factor?

For example, the expression \(7^7 * 13^3 * 17\) would have 17 as the greatest prime factor. Correct?


How else? Exponentiation does not "produce" primes: if p is a prime number then p^12 or p^10000 will still have only one prime - p.
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
G
mikemcgarry
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4493
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  05 Jun 2012, 14:22
4
Kudos
Expert Reply
Val1986 wrote:
What is the greatest prime factor of 4^17 - 2^28?

A. 2
B. 3
C. 5
D. 7
E. 11

Is there a shortcut to solving this question? :?


I'm happy to help with this. :)

We know 4 = 2^2, so 4^17 = (2^2)^17 = 2^(2*17) = 2^34

That takes advantage of a law of exponents that says (a^n)^m = a^(n*m)

So, 4^17 - 2^28 = 2^34 - 2^28 = 2^(28 + 6) - 2^28 = (2^28)*(2*6) - 2^28 = (2^6 - 1) *(2^28)
= (64 - 1)*(2^28) = 63*(2^28)

The prime factors of 63 are 3*3*7, so the largest prime factor is 7, answer choice D.

Here's a blog you may find helpful.
http://magoosh.com/gmat/2012/gmat-math-factors/

Does all that make sense? Please let me know if you have any further questions.

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)
Signature Read More
avatar
pike
Current Student
Joined: 08 Jan 2009
Posts: 266
GMAT 1: 770 Q50 V46
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  05 Jun 2012, 14:24
1
Kudos
4^17 - 2^28
= 2^34 - 2^28
= 2^28 * (2^6 - 1)
= (2*2*2*...) * (63)
= (2*2*2*...) * (3 * 3 * 7)

Greatest prime factor is 7
User avatar
B
thevenus
Senior Manager
Senior Manager
Joined: 17 Mar 2010
Status:Final Countdown
Posts: 372
Location: United States (NY)
GPA: 3.82
WE:Account Management (Retail Banking)
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  30 Aug 2012, 10:46
1
Kudos
Let's simplify (4)^17-2^28
={(2)^2}^17-(2)^28
=(2)^34-(2)^28
=(2)^28{(2)^6-1}
=(2)^28{64-1}
=(2)^28{63}
=(2)^28{3x3x7}

clearly 7
User avatar
S
Marcab
Director
Director
Joined: 03 Feb 2011
Status:Retaking after 7 years
Posts: 944
Location: United States (NY)
Concentration: Finance, Economics
Schools: Simon '15 (M$)
GMAT 1: 720 Q49 V39
GPA: 3.75
GMAT ToolKit User Premium Member
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  12 Nov 2012, 07:30
1
Kudos
1
Bookmarks
carcass wrote:
What is the greatest prime factor of \(4^17 - 2^28\)?

(A) 2
(B) 3
(C) 5
(D) 7
(E) 11


4^17 can be written as 2^34.
Hence we have to find out the greatest prime factor of 2^34-2^28.
Take 2^28 as common.
2^28(2^6-1)
It will become, 2^28 * (64-1)
2^28 * 63
Greatest prime factor of 2^28=2
Greatest prime factor of 63 is 7.
2^28 * 7*9
Therefore the answer is 7.
Hence D
_________________
//Retaking GMAT after 7 years//
Prepositional Phrases Clarified|Elimination of BEING| Absolute Phrases Clarified
Rules For Posting
Signature Read More
avatar
dbiersdo
Intern
Intern
Joined: 03 Aug 2012
Posts: 18
Location: United States (OR)
Concentration: Finance, International Business
Schools: Marshall PM '18 (I)
GPA: 3.53
WE:Analyst (Entertainment and Sports)
Reviews Badge
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  12 Nov 2012, 11:04
thevenus wrote:

=(2)^34-(2)^28
=(2)^28{(2)^6-1}


I don't get the transition here... what's the process involved that these two equal one another??
User avatar
V
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 77369
Premium Member CAT Tests Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  12 Nov 2012, 11:10
Expert Reply
dbiersdo wrote:
thevenus wrote:

=(2)^34-(2)^28
=(2)^28{(2)^6-1}


I don't get the transition here... what's the process involved that these two equal one another??


\(4^{17}-2^{28}=2^{34}-2^{28}\) --> factor out 2^28: \(2^{34}-2^{28}=2^{28}(2^6-1)=2^{28}*63=2^{28}*3^2*7\) --> the greatest prime factor is 7.

Hope it's clear.
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
V
BrentGMATPrepNow
GMAT Club Legend
GMAT Club Legend
Joined: 11 Sep 2015
Posts: 5493
Location: Canada
Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  19 Apr 2016, 17:17
Expert Reply
Flexxice wrote:
What is the greatest prime factor of 4^17 - 2^28?

A) 2

B) 3

C) 5

D) 7

E) 11



First of all 4^17 = (2^2)^17 = 2^34

So, 4^17 - 2^28 = 2^34 - 2^28
= 2^28(2^6 - 1) [factored out 2^28]
= 2^28(2^3 + 1)(2^3 - 1) [factored again]
= 2^28(8 + 1)(8 - 1) [evaluated]
= 2^28(9)(7) [evaluated]
= 2^28(3)(3)(7)

So, as you can see, the greatest prime factor is 7

Answer = D

Cheers,
Brent
_________________
Image

A focused approach to GMAT mastery
Signature Read More
avatar
Flexxice
Intern
Intern
Joined: 10 Mar 2016
Posts: 15
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  19 Apr 2016, 20:21
GMATPrepNow wrote:
Flexxice wrote:
What is the greatest prime factor of 4^17 - 2^28?

A) 2

B) 3

C) 5

D) 7

E) 11



First of all 4^17 = (2^2)^17 = 2^34

So, 4^17 - 2^28 = 2^34 - 2^28
= 2^28(2^6 - 1) [factored out 2^28]
= 2^28(2^3 + 1)(2^3 - 1) [factored again]
= 2^28(8 + 1)(8 - 1) [evaluated]
= 2^28(9)(7) [evaluated]
= 2^28(3)(3)(7)

So, as you can see, the greatest prime factor is 7

Answer = D

Cheers,
Brent



Thank you! I tried it in another way and I do not get why I got another answer than you did.

4^17 - 2^28 = 4^17 - 4^14 = 4^3 = 2^6

Because of that, I chose answer A.
User avatar
V
chetan2u
Math Expert
Joined: 02 Aug 2009
Posts: 10022
Reviews Badge Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  19 Apr 2016, 21:04
Expert Reply
Flexxice wrote:
GMATPrepNow wrote:
Flexxice wrote:
What is the greatest prime factor of 4^17 - 2^28?

A) 2

B) 3

C) 5

D) 7

E) 11



First of all 4^17 = (2^2)^17 = 2^34

So, 4^17 - 2^28 = 2^34 - 2^28
= 2^28(2^6 - 1) [factored out 2^28]
= 2^28(2^3 + 1)(2^3 - 1) [factored again]
= 2^28(8 + 1)(8 - 1) [evaluated]
= 2^28(9)(7) [evaluated]
= 2^28(3)(3)(7)

So, as you can see, the greatest prime factor is 7

Answer = D

Cheers,
Brent



Thank you! I tried it in another way and I do not get why I got another answer than you did.

4^17 - 2^28 = 4^17 - 4^14 = 4^3 = 2^6

Because of that, I chose answer A.


Hi,
you are wrong in the highlighted portion--
\(\frac{4^{17}}{4^{14}}= 4^{17-14} =4^3\)
\(4^{17} - 4^{14} = 4^{14}(4^3-1) = 4^{14} *(64-1) = 4^{14}*63= 4^{14}*7*9..\)
so 7 is the answer..
You can not subtract the way you have done, it can be done ONLY if two terms are being divided as shown above..

a simpler examplewill be
\(2^3 -2^1\).. as per you it will be\(2^2=4.\).
BUT \(2^3-2^1=8-2=6..\)
hope it helps
avatar
B
OptimusPrepJanielle
SVP
SVP
Joined: 06 Nov 2014
Posts: 1841
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  21 Apr 2016, 00:57
Expert Reply
Flexxice wrote:


Thank you! I tried it in another way and I do not get why I got another answer than you did.

4^17 - 2^28 = 4^17 - 4^14 = 4^3 = 2^6

Because of that, I chose answer A.


We cannot simply subtract the terms.

4^17 - 4^14 = 4^14 (4^3 - 1) = 4^14*(64 - 1) = 4^14 * 63
Here we have taken 4^14 common from both the terms and written the remainder inside the brackets.
User avatar
Aline2289
Intern
Intern
Joined: 31 Jan 2017
Posts: 4
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  15 Mar 2017, 16:40
Bunuel wrote:
student26 wrote:
What is the greatest prime factor of 4^17 - 2^28?

A. 2
B. 3
C. 5
D. 7
E. 11


\(4^{17}-2^{28}=2^{34}-2^{28}=2^{28}(2^6-1)=2^{28}*63=2^{28}*3^2*7\) --> greatest prime factor is 7.

Answer: D.



Why is it (2^6-1)? :oops:
User avatar
V
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 77369
Premium Member CAT Tests Top Member of the Month
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  16 Mar 2017, 06:07
1
Kudos
Expert Reply
Aline2289 wrote:
Bunuel wrote:
student26 wrote:
What is the greatest prime factor of 4^17 - 2^28?

A. 2
B. 3
C. 5
D. 7
E. 11


\(4^{17}-2^{28}=2^{34}-2^{28}=2^{28}(2^6-1)=2^{28}*63=2^{28}*3^2*7\) --> greatest prime factor is 7.

Answer: D.



Why is it (2^6-1)? :oops:


Added extra step:

\(4^{17}-2^{28}=2^{34}-2^{28}=2^{28}*2^6-2^{28}=2^{28}(2^6-1)=2^{28}*63=2^{28}*3^2*7\) --> greatest prime factor is 7.

So, we are factoring out \(2^{28}\) from \(2^{34}-2^{28}\).

Hope it's clear.
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
V
ScottTargetTestPrep
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 14313
Location: United States (CA)
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  21 Mar 2017, 05:10
1
Kudos
Expert Reply
student26 wrote:
What is the greatest prime factor of 4^17 - 2^28?

A. 2
B. 3
C. 5
D. 7
E. 11


We need to determine the greatest prime factor of 4^17 – 2^28. We can start by breaking 4^17 into prime factors.

4^17 = (2^2)^17 = 2^34

Now our equation is as follows:

2^34 – 2^28

Note that the common factor in each term is 2^28; thus, the expression can be simplified as follows:

2^28(2^6 – 1)

2^28(64 – 1)

2^28(63)

2^28 x 9 x 7

2^28 x 3^2 x 7

We see that the greatest prime factor must be 7.

Answer: D
_________________

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

Signature Read More
avatar
CrushGMAT
Intern
Intern
Joined: 20 Aug 2018
Posts: 24
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink] New post  24 Nov 2018, 14:18
A video explanation can be found here:
https://www.youtube.com/watch?v=gCL6DmaIqK4

The first step is noting that we want to work from a common base, i.e. instead of 4 and 2, we want 2^2 and 2. Now we have

(2^2)^17 - 2^28

What’s tested are your knowledge of exponent rules, factoring, and prime numbers. We have

2^34 – 2^28, which is

2^28(2^6 – 1) [note that 2^6 can be viewed more simply as 2^3*2^3, or 8*8], which is

2^28(64 – 1), which is

2^28 (63), which is

2^28 (3^2)(7)

Prime factors are 2, 3 and 7. Greatest prime factor is 7
_________________
http://www.crushthegmat.org
Signature Read More
GMAT Club Bot
gmatclubot
Re: What is the greatest prime factor of 4^17 - 2^28? [#permalink]
24 Nov 2018, 14:18
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
 1   2   
Moderators:
chetan2u
Math Expert
10022 posts
Bunuel
Math Expert
77369 posts
yashikaaggarwal
Senior Moderator - Masters Forum
2547 posts
generis
Senior SC Moderator
4820 posts

cron

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

Main navigation

GMAT Resources

Partners

MBA Resources

www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions