BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

What is the greatest prime factor of 3^6 - 1?

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

by Jay@ManhattanReview » Mon Dec 11, 2017 7:38 am
swerve wrote:What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17

The OA is D.

Please, can any expert explain this PS question for me? I don't understand why that is the correct answer. Thanks.
3^6 - 1 = 3^6 - 1^6 = (3^3)^2 - (1^3)^2 = (3^3 - 1^3) * (3^3 + 1^3); applying a^2 - b^2 = (a - b)(a + b)

(3^3 - 1^3) * (3^3 + 1^3) = [(3 - 1)(3^2 + 3*1 + 1^2)]*[(3 + 1)(3^2 - 3*1 + 1^2)] = 2*(9 + 3 =1)*4*(9 - 3 +1) = 2*13*4*7

We see that the greatest prime factor is 13.

The correct answer: D

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: Stamford | Istanbul | Evanston | Seoul | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Mon Dec 11, 2017 7:38 am
swerve wrote:What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17
3^6 - 1 is a difference of squares, since 3^6 = (3³)² and 1 = 1²

So, we get: 3^6 - 1 = (3³ + 1)(3³ - 1)
= (27 + 1)(27 - 1)
= (28)(26)
= (2)(2)(7)(2)(13)

The prime factors are 2, 7 and 13
The greatest value is 13

Answer: D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

by Jay@ManhattanReview » Mon Dec 11, 2017 7:42 am
swerve wrote:What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17

The OA is D.

Please, can any expert explain this PS question for me? I don't understand why that is the correct answer. Thanks.
You may use brute force to get this one sorted.

We have 3^6 - 1 = 729 - 1 = 728

Since we want the greatest prime factor and the five options are arranged in an ascending order, start dividing 728 one by one starting from option E onwards in the reverse order.

We see that 728 is not divisible by 17 but is divisible by 13, so 13 is the correct answer.

The correct answer: D

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: Stamford | Istanbul | Evanston | Seoul | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
Top

GMAT/MBA Expert

User avatar
Legendary Member
Posts: 503
Joined: Thu Jul 20, 2017 9:03 am
Thanked: 86 times
Followed by:15 members
GMAT Score:770

by ErikaPrepScholar » Mon Dec 11, 2017 8:01 am
3^6 is a little too large of an exponent to deal with in our heads, so we'll try to factor this out. We'll start by expressing $$3^6-1$$ as $$ \left(3^3-1\right)\left(3^3+1\right)$$ 3^3 is 27, so this gives $$\left(27-1\right)\left(27+1\right)=26\cdot28$$
Then we can do the prime factorization of each number:
$$26\cdot28=\left(2\cdot13\right)\left(2\cdot2\cdot7\right)$$
So the prime factors are 2, 7, and 13, giving a greatest prime factor of 13.
Image

Erika John - Content Manager/Lead Instructor
https://gmat.prepscholar.com/gmat/s/

Get tutoring from me or another PrepScholar GMAT expert: https://gmat.prepscholar.com/gmat/s/tutoring/

Learn about our exclusive savings for BTG members (up to 25% off) and our 5 day free trial

Check out our PrepScholar GMAT YouTube channel, and read our expert guides on the PrepScholar GMAT blog
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 8086
Joined: Sat Apr 25, 2015 10:56 am
Location: Los Angeles, CA
Thanked: 43 times
Followed by:29 members

by Scott@TargetTestPrep » Wed Jan 30, 2019 5:58 pm
swerve wrote:What is the greatest prime factor of 3^6 - 1 ?

A. 2
B. 3
C. 7
D. 13
E. 17
We can factor 3^6 - 1 as a difference of squares. Factoring gives us:

3^6 - 1 = (3^3 - 1)(3^3 + 1) = 26 x 28 = 13 x 2 x 7 x 4, so the greatest prime factor is 13.

Answer: D

Scott Woodbury-Stewart
Founder and CEO
[email protected]

Image

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

ImageImage
Top

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Wed Jan 30, 2019 7:09 pm
Hi All,

A number of the posts have provided elegant solutions to this question. The basic math behind this prompt is Arithmetic and Prime Factorization though, so if you don't immediately "see" the elegant approach, you can still get to the answer....

We're asked to find the LARGEST prime factor of 3^6 - 1

3^6 = 9^3 = (9)(9)(9) = 729

729 - 1 = 728

Now, we can prime factor 728.

You probably immediately see that 728 is divisible by 2, but if you know your 'rules of division', you can see that it's also divisible by 4...

728 =
(4)(182)
(4)(2)(91)
(4)(2)(7)(13)

13 is the largest prime.

Sometimes this type of approach isn't practical (especially if the numbers involved are HUGE), but when you're given 'manageable' numbers, there's nothing wrong with admitting that you don't see the 'hidden pattern.' If you can get to the correct answer in a reasonable amount of time by just doing arithmetic, then it's better to do THAT than waste time staring at the screen.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
Image
Top
Post new topic Post Reply
• Page 1 of 1