• 5-Day Free Trial
5-day free, full-access trial TTP Quant

Available with Beat the GMAT members only code

• Free Trial & Practice Exam
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Free Practice Test & Review
How would you score if you took the GMAT

Available with Beat the GMAT members only code

• Get 300+ Practice Questions

Available with Beat the GMAT members only code

• Free Veritas GMAT Class
Experience Lesson 1 Live Free

Available with Beat the GMAT members only code

• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

• Award-winning private GMAT tutoring
Register now and save up to $200 Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • 1 Hour Free BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code ## For the positive integers a, b (OG16) tagged by: rolandprowess This topic has 1 expert reply and 1 member reply boomgoesthegmat Senior | Next Rank: 100 Posts Joined 25 Apr 2016 Posted: 93 messages Thanked: 1 times #### For the positive integers a, b (OG16) Thu May 19, 2016 3:43 pm Elapsed Time: 00:00 • Lap #[LAPCOUNT] ([LAPTIME]) 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 Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums! OptimusPrep Master | Next Rank: 500 Posts Joined 13 Mar 2015 Posted: 410 messages Followed by: 7 members Thanked: 120 times GMAT Score: 770 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 Prepâ„¢ Optimus Prepâ„¢ 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 Prepâ„¢ 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

### GMAT/MBA Expert

ceilidh.erickson GMAT Instructor
Joined
04 Dec 2012
Posted:
1706 messages
Followed by:
224 members
Thanked:
1443 times
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

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!

Free Manhattan Prep online events - The first class of every online Manhattan Prep course is free. Classes start every week.

### Best Conversation Starters

1 lheiannie07 116 topics
2 ardz24 64 topics
3 swerve 63 topics
4 LUANDATO 62 topics
5 M7MBA 57 topics
See More Top Beat The GMAT Members...

### Most Active Experts

1 Brent@GMATPrepNow

GMAT Prep Now Teacher

170 posts
2 EconomistGMATTutor

The Economist GMAT Tutor

129 posts
3 Rich.C@EMPOWERgma...

EMPOWERgmat

122 posts
4 GMATGuruNY

The Princeton Review Teacher

121 posts
5 Scott@TargetTestPrep

Target Test Prep

118 posts
See More Top Beat The GMAT Experts