• Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

    MORE DETAILS
    Veritas Prep
  • examPAL
    Most awarded test prep in the world
    Now free for 30 days

    Available with Beat the GMAT members only code

    MORE DETAILS
    examPAL
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • Target Test Prep
    5-Day Free Trial
    5-day free, full-access trial TTP Quant

    Available with Beat the GMAT members only code

    MORE DETAILS
    Target Test Prep
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider
  • Varsity Tutors
    Award-winning private GMAT tutoring
    Register now and save up to $200

    Available with Beat the GMAT members only code

    MORE DETAILS
    Varsity Tutors
  • Economist Test Prep
    Free Trial & Practice Exam
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep
  • e-gmat Exclusive Offer
    Get 300+ Practice Questions
    25 Video lessons and 6 Webinars for FREE

    Available with Beat the GMAT members only code

    MORE DETAILS
    e-gmat Exclusive Offer
  • PrepScholar GMAT
    5 Day FREE Trial
    Study Smarter, Not Harder

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • Kaplan Test Prep
    Free Practice Test & Review
    How would you score if you took the GMAT

    Available with Beat the GMAT members only code

    MORE DETAILS
    Kaplan Test Prep

For the positive integers a, b (OG16)

This topic has 1 expert reply and 1 member reply
boomgoesthegmat Senior | Next Rank: 100 Posts Default Avatar
Joined
25 Apr 2016
Posted:
93 messages
Upvotes:
1

For the positive integers a, b (OG16)

Post Thu May 19, 2016 3:43 pm
For the positive integers a, b, and k, a^k || b means that a^k is a divisor of b, but a^(k+1) is not a divisor of b. If k is a positive integer and 2^k ||72, then k is equal to

A) 2
B) 3
C) 4
D) 8
E) 18

OA: B

  • +1 Upvote Post
  • Quote
  • Flag
Post Mon Oct 09, 2017 7:57 am
Any positive integer can be expressed as the product of prime factors - we call this PRIME FACTORIZATION. For example, the prime factorization of 140 is (2^2)(5^1)(7^1), and 108 = (2^2)(3^3). This prime factorization is really helpful when you're solving questions with variable exponents. For example:

12 = (2^x)(3^y)
In order to solve for x and y here, we need to break 12 down to its prime factors:
12 = (2^2)(3)
So, (2^2)(3) = (2^x)(3^y)
Therefore x = 2 and y = 1.

In #110, we have "k" and "k + 1" as exponents, which is a big clue that we probably want to think in terms of prime factorization.

The hardest part about this question is simply figuring out what it's asking - breaking down "a^k is a divisor of b, but a^(k + 1) is not." How do we translate that? So "a" to a certain exponent goes evenly into "b," but "a" to the next highest exponent does not. That would mean that a^k was the maximum number of a's that go into "b." In other words... all the a's!

So, 2^k || 72 would be the maximum number of factors of 2 that go into 72 - all the 2's in 72. If we factor 72, we find that the prime factorization is (2^3)(3^2). There are 3 factors of 2 in 72 (that's what 2^3 tells us), so k must equal 3.

There's more on prime factorization here: http://www.beatthegmat.com/2-x-2-x-2-3-2-13-what-is-x-t178975.html#578272

_________________


Ceilidh Erickson
Manhattan Prep GMAT & GRE instructor
EdM in Mind, Brain, and Education
Harvard Graduate School of Education


Manhattan Prep instructors all have 99th+ percentile scores and expert teaching experience.
Sign up for a FREE TRIAL, and learn why we have the highest ratings in the GMAT industry!

  • +1 Upvote Post
  • Quote
  • Flag
Free Manhattan Prep online events - The first class of every online Manhattan Prep course is free. Classes start every week.
OptimusPrep Master | Next Rank: 500 Posts
Joined
13 Mar 2015
Posted:
410 messages
Followed by:
7 members
Upvotes:
120
GMAT Score:
770
Post Thu May 19, 2016 7:58 pm
boomgoesthegmat wrote:
For the positive integers a, b, and k, a^k || b means that a^k is a divisor of b, but a^(k+1) is not a divisor of b. If k is a positive integer and 2^k ||72, then k is equal to

A) 2
B) 3
C) 4
D) 8
E) 18

OA: B
2^k ||72 means 2^k is a divisor of 72, but 2^(k+1) is not a divisor of 72

72 = 2^3*3^2

The maximum powers of 2 in 72 = 3
Hence 2^3 is a divisor of 72 and 2^4 is not a divisor of 72
k = 3

Correct Option: B

_________________
Ankur
99th Percentile GMAT Tutor | Optimus Prep™

Optimus Prep™ GMAT Courses:
www.Optimus-Prep.com/GMAT

Free Online Trial Hour:
http://www.optimus-prep.com/request-free-online-trial-hour/

Free GMAT Study Plan:
http://www.optimus-prep.com/ebook/

Optimus Prep™ Rates:
GMAT Private Tutoring Online: $80-100/hr.*
GMAT Private Tutoring In-Person: $130-150/hr.*
GMAT On Demand Course: $299 (Use Discount Code BEATTHEGMAT111)*

*All GMAT Courses Have A 50 Points Score Improvement Or Full Refund Guarantee

  • +1 Upvote Post
  • Quote
  • Flag

Best Conversation Starters

1 lheiannie07 112 topics
2 ardz24 71 topics
3 Roland2rule 69 topics
4 LUANDATO 53 topics
5 swerve 45 topics
See More Top Beat The GMAT Members...

Most Active Experts

1 image description GMATGuruNY

The Princeton Review Teacher

154 posts
2 image description Rich.C@EMPOWERgma...

EMPOWERgmat

107 posts
3 image description Jeff@TargetTestPrep

Target Test Prep

106 posts
4 image description Scott@TargetTestPrep

Target Test Prep

98 posts
5 image description EconomistGMATTutor

The Economist GMAT Tutor

91 posts
See More Top Beat The GMAT Experts