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

Redeem
TTP OnDemand GMAT Video Course
Featured Course

Target Test Prep GMAT OnDemand

Scott Woodbury-Stewart’s private virtual classroom — 400 hours of master-class video lessons for the GMAT Focus Edition.

  • 715+ score guarantee — highest in the industry (99th percentile)
  • 52 chapters · 1,500+ lessons · 4,000+ practice questions
  • 400 hours of video · 1,500+ instructor-led HD smartboard lessons
  • 300,000+ students accepted to Harvard, Stanford, Wharton, Booth & Sloan & more
  • 24/7 live support + weekly Zoom office hours with GMAT instructors
  • TTP AI Assist — 24/7 AI-powered virtual tutor for instant help
  • 1,200+ flashcards + AI-powered study assistant & daily calendar
  • OnDemand, LiveTeach & GMAT Bootcamp formats available
  • Also: GRE, SAT Math & Executive Assessment courses
  • MBA Admissions Consulting now available
  • 🏆 2025 EdTech Breakthrough Award: Test Prep Solution Provider of the Year
  • 200,000+ students served
  • 5-day free trial — $0 to start, no auto-billing, cancel anytime
★★★★★ 5.0 (559 reviews)
130-pt guarantee
$0 to start then $127/mo
Start Free 5-Day Trial Read 559 Reviews →

If 5400mn = k^4

Expert replies
Post new topic Post Reply
Previous Topic Next Topic
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3835
Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members
Location: Milpitas, CA
Thanked: 1854 times
GMAT Score:770

by Anurag@Gurome » Sun Feb 27, 2011 3:48 am
Baten80 wrote:If 5400mn = k^4, where m, n, and k are positive integers, what is the least possible value of m + n?
A. 11
B. 18
C. 20
D. 25
E. 33
5400 = 2^3 * 3^3 * 5^2
5400mn = 2^3 * 3^3 * 5^2 * mn
Since 5400mn = k^4, so 2, 3, and 5 should be all 4th powers
2^3 * 3^3 * 5^2 * mn = k^4 implies mn = 2 * 3 * (5^2) = 150, which is the least value of mn.
15 = 15 * 10 (15 and 10 are the least possible factors of 150)
Now to find the least possible value of m + n = 15 + 10 = 25

The correct answer is D.
Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

Join Our Facebook Groups
GMAT with Gurome
https://www.facebook.com/groups/272466352793633/
Admissions with Gurome
https://www.facebook.com/groups/461459690536574/
Career Advising with Gurome
https://www.facebook.com/groups/360435787349781/
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Senior | Next Rank: 100 Posts
Posts: 45
Joined: Sun Oct 16, 2011 2:21 pmFollowed by: 1 member

by lenagmat » Tue Oct 25, 2011 7:21 am
Please could you explain how does it implies mn = 2 * 3 * (5^2) = 150 from 2^3 * 3^3 * 5^2 * mn = k^4

Anurag@Gurome wrote:
Baten80 wrote: Since 5400mn = k^4, so 2, 3, and 5 should be all 4th powers
2^3 * 3^3 * 5^2 * mn = k^4 implies mn = 2 * 3 * (5^2) = 150, which is the least value of mn.
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Senior | Next Rank: 100 Posts
Posts: 44
Joined: Wed Aug 25, 2010 12:23 am
Thanked: 6 times

by rooster » Tue Oct 25, 2011 9:14 pm
lenagmat wrote:Please could you explain how does it implies mn = 2 * 3 * (5^2) = 150 from 2^3 * 3^3 * 5^2 * mn = k^4

Anurag@Gurome wrote:
Baten80 wrote: Since 5400mn = k^4, so 2, 3, and 5 should be all 4th powers
2^3 * 3^3 * 5^2 * mn = k^4 implies mn = 2 * 3 * (5^2) = 150, which is the least value of mn.
There are a few hints that will let us know. First you need to break this into primes

5400mn=k^4 and k is a number

so that means the fourth root of 5400mn is a number

5400mn= (3*3*3*2*2*2*5*5)mn

In order for this to be a perfect fourth root you need 3*2*5*5 which means mn needs to have a combination of those 4 numbers.

15 +10 satisfies this, therefor the answer is D
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
User avatar
GMAT Instructor
Posts: 349
Joined: Wed Sep 28, 2011 3:38 pmFollowed by: 54 members
Location: Austin, TX
Thanked: 236 times
GMAT Score:770

by GmatMathPro » Wed Oct 26, 2011 6:07 am
lenagmat wrote:Please could you explain how does it implies mn = 2 * 3 * (5^2) = 150 from 2^3 * 3^3 * 5^2 * mn = k^4
By itself it doesn't imply that exactly. The idea is that 5400mn=k^4 implies that 5400mn can be represented as some integer raised to the power of 4. For that to be true, all of the exponents in the number's prime factorization must be multiples of 4. For example, 2^12*3^8*5^4=(2^3*3^2*5)^4=360^4. This is only possible because the exponents in the prime factorization(12,8,4) are all multiples of 4.

In this problem, 5400mn=2^3*3^3*5^2mn, so we need to choose mn so that it makes the exponents on the right side all multiples of 4. If mn=2*3*5^2, then the expression becomes 2^4*3^4*5^4=(2*3*5)^4=30^4. However, if we are only interested in making it a 4th power, we also could choose mn=2*3^5*5^6, so the expression becomes 2^4*3^8*5^8=(2*3^2*5^2)^4=450^4. We have infinitely many choices for mn to make 5400mn equal to an integer raised to the fourth power. However, in this problem we also want to minimize m+n which implies that we should choose the lowest possible value of mn.
Pete Ackley
GMAT Math Pro
Free Online Tutoring Trial
Top
Join the discussion
Log in to postLogin with Googleuse password
QUOTE IN REPLY
Post new topic Post Reply
• Page 1 of 1