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

 It is currently 16 Oct 2018, 10:35

### 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

# What is the greatest prime factor of 4^17 - 2^28?

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

### Hide Tags

Intern
Joined: 25 Oct 2010
Posts: 32
WE 1: 3 yrs
What is the greatest prime factor of 4^17 - 2^28?  [#permalink]

### Show Tags

13 Nov 2010, 10:55
6
45
00:00

Difficulty:

25% (medium)

Question Stats:

69% (00:35) correct 31% (00:38) wrong based on 1824 sessions

### HideShow timer Statistics

What is the greatest prime factor of 4^17 - 2^28?

A. 2
B. 3
C. 5
D. 7
E. 11
Math Expert
Joined: 02 Sep 2009
Posts: 49915

### Show Tags

13 Nov 2010, 11:03
11
21
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.

_________________
##### General Discussion
Manager
Status: Planning to retake.
Affiliations: Alpha Psi Omega
Joined: 25 Oct 2010
Posts: 85
Concentration: General Management, Entrepreneurship
GMAT 1: 650 Q42 V37
GRE 1: Q630 V680
GPA: 3.16

### Show Tags

13 Nov 2010, 12:40
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?
_________________

Each moment of time ought to be put to proper use, either in business, in improving the mind, in the innocent and necessary relaxations and entertainments of life, or in the care of the moral and religious part of our nature.

-William Andrus Alcott

Math Expert
Joined: 02 Sep 2009
Posts: 49915

### Show Tags

13 Nov 2010, 12:47
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).
_________________
Manager
Status: Planning to retake.
Affiliations: Alpha Psi Omega
Joined: 25 Oct 2010
Posts: 85
Concentration: General Management, Entrepreneurship
GMAT 1: 650 Q42 V37
GRE 1: Q630 V680
GPA: 3.16

### Show Tags

13 Nov 2010, 13:05
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?
_________________

Each moment of time ought to be put to proper use, either in business, in improving the mind, in the innocent and necessary relaxations and entertainments of life, or in the care of the moral and religious part of our nature.

-William Andrus Alcott

Math Expert
Joined: 02 Sep 2009
Posts: 49915

### Show Tags

13 Nov 2010, 13:10
1
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.
_________________
Manager
Joined: 08 Jun 2010
Posts: 109

### Show Tags

17 Jan 2011, 10:01
what A silly mistake.
I have solved it ,but in last I assume that I have to find the greatest n.m. of prime factors.not the n.m. itself
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4494
Re: what is the greatest prime factor?  [#permalink]

### Show Tags

05 Jun 2012, 15:22
3
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)

Current Student
Joined: 08 Jan 2009
Posts: 304
GMAT 1: 770 Q50 V46
Re: what is the greatest prime factor?  [#permalink]

### Show Tags

05 Jun 2012, 15:24
1
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
Manager
Status: Rising GMAT Star
Joined: 05 Jun 2012
Posts: 122
Location: Philippines
Concentration: General Management, Finance
GPA: 3.22
WE: Corporate Finance (Consulting)
Re: what is the greatest prime factor?  [#permalink]

### Show Tags

05 Jun 2012, 18:58
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?

Answered this in 40 seconds (is that already considered a short cut? hehe)

= 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)

Now here comes the intuitive part
2^28 = greatest prime factor is 2
63 = 9 * 7 = 3 * 3 * 7 (prime factorization) = greatest prime factor is 7

_________________

Far better is it to dare mighty things, to win glorious triumphs, even though checkered by failure... than to rank with those poor spirits who neither enjoy nor suffer much, because they live in a gray twilight that knows not victory nor defeat.
- T. Roosevelt

Senior Manager
Joined: 13 Jan 2012
Posts: 286
Weight: 170lbs
GMAT 1: 740 Q48 V42
GMAT 2: 760 Q50 V42
WE: Analyst (Other)
Re: what is the greatest prime factor?  [#permalink]

### Show Tags

06 Jun 2012, 00:41
1
mikemcgarry wrote:
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

Wow. I am floored by how great of an explanation you provided. Posts like that make me really think that doing thousands of practice problems with good explanations beats out reading books on math every day of the week.
Senior Manager
Status: Final Countdown
Joined: 17 Mar 2010
Posts: 460
Location: India
GPA: 3.82
WE: Account Management (Retail Banking)
What is the greatest prime factor of  [#permalink]

### Show Tags

30 Aug 2012, 11:46
1
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
_________________

" Make more efforts "
Press Kudos if you liked my post

Senior Manager
Joined: 03 Sep 2012
Posts: 382
Location: United States
Concentration: Healthcare, Strategy
GMAT 1: 730 Q48 V42
GPA: 3.88
WE: Medicine and Health (Health Care)
Re: What is the greatest prime factor of 4^17 - 2^28?  [#permalink]

### Show Tags

01 Oct 2012, 06:03
4^17 – 2^28
4^17 can be written as (2^2)^17 = 234
Therefore we get 2^34 - 2^28
Taking 22^8 as common we get 22^8 (26 – 1)

2^28 x 63

2^28 has only one prime factor ie. 2
63 has 3 x 3 x 7 as its prime factors …

Therefore all the factors of this product would be 2^28 x 3 x 3 x 7 ..

Thus 7 is the greatest factor (D)
_________________

"When you want to succeed as bad as you want to breathe, then you’ll be successful.” - Eric Thomas

VP
Status: Been a long time guys...
Joined: 03 Feb 2011
Posts: 1170
Location: United States (NY)
Concentration: Finance, Marketing
GPA: 3.75
Re: What is the greatest prime factor of 4^17 - 2^28  [#permalink]

### Show Tags

12 Nov 2012, 08:30
1
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
_________________
Intern
Joined: 03 Aug 2012
Posts: 19
Location: United States (OR)
Concentration: Finance, International Business
GPA: 3.53
WE: Analyst (Entertainment and Sports)
Re: What is the greatest prime factor of  [#permalink]

### Show Tags

12 Nov 2012, 12:04
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??
Math Expert
Joined: 02 Sep 2009
Posts: 49915
Re: What is the greatest prime factor of  [#permalink]

### Show Tags

12 Nov 2012, 12:10
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.
_________________
Board of Directors
Joined: 01 Sep 2010
Posts: 3305
Re: What is the greatest prime factor of 4^17 - 2^28?  [#permalink]

### Show Tags

12 Nov 2012, 17:37
Great explanation Mod

Those questions seems simple when you master the concepts but indeed are really tough

Quote:
Wow. I am floored by how great of an explanation you provided. Posts like that make me really think that doing thousands of practice problems with good explanations beats out reading books on math every day of the week.

when you master the concept and you know them cold..........In my opinion the only way is to practice questions from all level to see different things from odds angles

_________________
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1829
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: What is the greatest prime factor of 4^17 - 2^28?  [#permalink]

### Show Tags

26 Feb 2014, 01:39
4^17 - 2^28 =
4^17 - 4^14 =
4^14 . (4^3 - 1) =

4^14 . 63 = 4^14. 9. 7

Answer = 7 = D
_________________

Kindly press "+1 Kudos" to appreciate

Manager
Joined: 05 Jun 2012
Posts: 74
Schools: IIMA
Re: What is the greatest prime factor of 4^17 - 2^28?  [#permalink]

### Show Tags

01 Jul 2014, 06:40
4 is not prime, so break the 4's down into 2's:
4^17 = (2^2)^17 = 2^34
so we have
2^34 - 2^28
at this point, make a common exponent, so that we can factor out the largest possible common factor.
(2^28)(2^6) - (2^28)
(2^28)(2^6 - 1)
(2^28)(63)
finish breaking into primes:
(2^28)(3)(3)(7)
so the greatest prime factor is 7
_________________

If you are not over prepared then you are under prepared !!!

Current Student
Joined: 12 Aug 2015
Posts: 2638
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: What is the greatest prime factor of 4^17 - 2^28?  [#permalink]

### Show Tags

14 Mar 2016, 08:21
The Key to all of these questions is => factorise and simplify
here taking 2^28 common and 2 is a prime factor
so we need to only factorise 63
hence 7 is the answer
_________________

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 GMAT-Quant(700+)

AVERAGE GRE Scores At The Top Business Schools!

Re: What is the greatest prime factor of 4^17 - 2^28? &nbs [#permalink] 14 Mar 2016, 08:21

Go to page    1   2    Next  [ 31 posts ]

Display posts from previous: Sort by

# What is the greatest prime factor of 4^17 - 2^28?

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

 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®.