• NEW! FREE Beat The GMAT Quizzes
    NEW! FREE Beat The GMAT Quizzes
    NEW! FREE Beat The GMAT Quizzes
    Hundreds of Questions Highly Detailed Reporting Expert Explanations TAKE A FREE GMAT QUIZ
  • 7 CATs FREE!
    If you earn 100 Forum Points

    Engage in the Beat The GMAT forums to earn
    100 points for $49 worth of Veritas practice GMATs FREE

    Veritas Prep
    VERITAS PRACTICE GMAT EXAMS
    Earn 10 Points Per Post
    Earn 10 Points Per Thanks
    Earn 10 Points Per Upvote
    REDEEM NOW

e-GMAT Question of the Week #1

This topic has 1 expert reply and 0 member replies

GMAT/MBA Expert

e-GMAT Question of the Week #1

Post

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

Difficult



The number n is the product of the first 49 natural numbers. What is the maximum possible value of p + q such that both $$\frac{n}{24^p}$$ and $$\frac{n}{36^q}$$ are integers?

A. 11
B. 15
C. 20
D. 26
E. 30


To access all questions: Consolidated list of Questions of the Week

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

GMAT Instructor
Joined
26 Jul 2010
Posted:
645 messages
Followed by:
225 members
Upvotes:
527
Post
Solution
Given:
    The number n is the product of the first 49 natural numbers
    The numbers $$\frac{n}{24^p}$$ and $$\frac{n}{36^q}$$ are integers

To find:
    The maximum possible value of p + q

Approach and Working:
    When it is given that $$\frac{n}{24^p}$$ is an integer, it necessarily means p is the highest power of 24 that can divide the number n
    In a similar way we can say that, if $$\frac{n}{36^q}$$ is an integer, then q is the highest power of 36 that can divide the number n

Now, n is defined as the product of the first 49 natural numbers - means n = 49!

So, effectively we are trying to find out the highest power of 24 (which is p) and 36 (which is q) respectively which can divide 49!
    Now, as the numbers 24 and 36 are composite numbers, to find the highest power of them, we need to express them in terms of the prime factors and then figure out the individual instances of those prime factors.


If we factorise the numbers 24 and 36, we get
    24 = 2^3 x 3^1
    36 = 2^2 x 3^2

As the numbers 24 and 36 both consist of powers of 2 and 3 only, first we will find out the instances of 2 and 3 individually in 49!
    The number of 2s present in 49! = 49/2 + 49/22 + 49/23 + 49/24 + 49/25 = 24 + 12 + 6 + 3 + 1 = 46
    The number of 3s present in 49! = 49/3 + 49/32 + 49/33 = 16 + 5 + 1 = 22

Considering the number 24, as it is equal to 2^3 x 3^1
    Number of 23s present = 46/3 = 15
    Number of 31s present = 22/1 = 22
    Hence, number of combinations possible for 2^3 - 3^1 =15

Therefore, we can say highest power of 24 present in 49! = max (p) = 15
In a similar way, considering the number 36, as it is equal to 2^2 x 3^2
    Number of 22s present = 46/2 = 23
    Number of 32s present = 22/2 = 11
    Hence, number of combinations possible for 2^2 x 3^2 =11

Therefore, we can say highest power of 36 present in 49! = max (q) = 11
As, we have the maximum values of p and q respectively, we can say
    Max (p + q) = 15 + 11 = 26

Hence, the correct answer is option D.
Answer: D

  • +1 Upvote Post
  • Quote
  • Flag
  • 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
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider
  • 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
  • Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

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

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep
  • 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
  • The Princeton Review
    FREE GMAT Exam
    Know how you'd score today for $0

    Available with Beat the GMAT members only code

    MORE DETAILS
    The Princeton Review
  • 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

Top First Responders*

1 Brent@GMATPrepNow 39 first replies
2 Ian Stewart 37 first replies
3 Jay@ManhattanReview 32 first replies
4 GMATGuruNY 26 first replies
5 Scott@TargetTestPrep 17 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

Most Active Experts

1 image description Scott@TargetTestPrep

Target Test Prep

188 posts
2 image description Max@Math Revolution

Math Revolution

84 posts
3 image description Brent@GMATPrepNow

GMAT Prep Now Teacher

71 posts
4 image description Ian Stewart

GMATiX Teacher

45 posts
5 image description GMATGuruNY

The Princeton Review Teacher

45 posts
See More Top Beat The GMAT Experts