NEW HERE? LEARN MORE
closeclose
Last visit was: 26 Apr 2024, 00:52 It is currently 26 Apr 2024, 00:52
Decision Tracker
My Rewards
New posts
New comers' posts

Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

m is the smallest positive integer such that for any integer n ≥ m

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92920
Own Kudos [?]: 619084 [13]
Given Kudos: 81596
Send PM
CAT Tests Forum Quiz Mode
m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   14 Feb 2020, 05:32
1
Kudos
10
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

95% (hard)

Question Stats:

39% (02:40) correct 61% (02:33) wrong based on 221 sessions
Hide Show timer Statistics
\(m\) is the smallest positive integer such that for any integer \(n≥m\), the quantity \(n^3–7n^2+11n–5\) is positive. What is the value of \(m\)?

A. 4
B. 5
C. 8
D. 6
E. 11


Are You Up For the Challenge: 700 Level Questions
ShowHide Answer
Official Answer
D
Most Helpful Reply
User avatar
S
dips1122
Manager
Manager
Joined: 12 Oct 2019
Posts: 76
Own Kudos [?]: 39 [6]
Given Kudos: 92
Location: India
Concentration: Marketing, General Management
Schools: ISB'22 (A) Tuck '23 (A$) Ross '23 (WL) IIMA PGPX'22 (WA) Fuqua '23 (WL) Darden '23 (WD) Cambridge '22 (D)
GMAT 1: 720 Q48 V41
GMAT 2: 730 Q50 V39
GMAT 3: 760 Q50 V44
GPA: 4
WE:Information Technology (Computer Software)
Send PM
Reviews Badge
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   14 Feb 2020, 17:59
6
Kudos
Quickly check for options. If we substitute n for 5, the equation = 0. But we need it to be +ve for all n>=m. At m=5, the value is 0. So for m>5, it will be positive. Smallest value of m = 6. IMO D
General Discussion
User avatar
L
shameekv1989
Director
Director
Joined: 14 Dec 2019
Posts: 829
Own Kudos [?]: 889 [0]
Given Kudos: 354
Location: Poland
Concentration: Entrepreneurship, Strategy
Schools: Cambridge '22 (D) WBS (A$) IIMA PGPX'22 (D) WHU MBA (WD) ISB'22 (D)
GMAT 1: 640 Q49 V27
GMAT 2: 660 Q49 V31
GMAT 3: 720 Q50 V38
GPA: 4
WE:Engineering (Consumer Electronics)
Send PM
GMAT ToolKit User Reviews Badge
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   14 Feb 2020, 06:52
Bunuel wrote:
\(m\) is the smallest positive integer such that for any integer \(n≥m\), the quantity \(n^3–7n^2+11n–5\) is positive. What is the value of \(m\)?

A. 4
B. 5
C. 8
D. 6
E. 11


The equation \(n^3–7n^2+11n–5\) = 0 at n = 5

Therefore for the expression to be positive n>5.
Thus the smallest positive integer for which \(n >= m\) m has to be 5

Answer - B
User avatar
L
Archit3110
GMAT Club Legend
GMAT Club Legend
Joined: 18 Aug 2017
Status:You learn more from failure than from success.
Posts: 8020
Own Kudos [?]: 4098 [0]
Given Kudos: 242
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1:
545 Q79 V79 DI73
GPA: 4
WE:Marketing (Energy and Utilities)
Send PM
GMAT ToolKit User CAT Tests
m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   14 Feb 2020, 23:44
re write given expression as
\(n^3–7n^2+11n–5\)
\((n^2(n-7)+11n–5)\)

plugin values of options since \(n≥m\)
at a=4 we get -ve
at a=5 we get 0
at a=6 we get +ve value
IMO D; sufficient


Bunuel wrote:
\(m\) is the smallest positive integer such that for any integer \(n≥m\), the quantity \(n^3–7n^2+11n–5\) is positive. What is the value of \(m\)?

A. 4
B. 5
C. 8
D. 6
E. 11


Are You Up For the Challenge: 700 Level Questions
avatar
B
Arvind90
Intern
Intern
Joined: 27 Jan 2016
Posts: 24
Own Kudos [?]: 13 [1]
Given Kudos: 53
Location: India
GMAT 1: 700 Q50 V38
GPA: 4
Send PM
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   07 Sep 2020, 16:45
1
Bookmarks
why cant m be any positive integer value less than 6? the equation states n>=m.
We know that the smallest value of n is 6 as the cubic equation is positive for n=6.

n is greater than or equal to m.

so then, m can be 5 or 4?

Can anyone explain what am i missing here?
User avatar
B
Dream2020
Intern
Intern
Joined: 07 Aug 2018
Status:Wings of Fire
Posts: 11
Own Kudos [?]: 10 [0]
Given Kudos: 47
Location: India
Concentration: Finance, General Management
Schools: ISB'22 NUS '22 IIMB EPGP'22 S.P.Jain Institute
GMAT 1: 610 Q47 V27
GMAT 2: 700 Q46 V40
WE:Accounting (Retail Banking)
Send PM
Reviews Badge
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   07 Sep 2020, 17:12
Is there any other way to solve this problem without resorting to Trail and error? It is very time consuming.
avatar
B
DippyDiva
Intern
Intern
Joined: 25 Dec 2019
Posts: 7
Own Kudos [?]: 0 [0]
Given Kudos: 9
Location: India
Schools: Booth '24 Darden '24 ESADE Foster '24 Fuqua '24 Haas '24 IESE '24 ISB '23 Judge '23 Kelley '24 Kellogg '24 LBS '24 NTU NUS Said '23 Bocconi Sloan '24 Tepper '24 Tuck '24 Yale '24
GMAT 1: 710 Q49 V34
GPA: 4
Send PM
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   08 Sep 2020, 10:55
Factorize the cubic equation to (n-1)(n-1)(n-5) which is positive for all integers greater than 5. So the smalleo value of n turns out to be 6 which equals m

Posted from my mobile device
User avatar
S
Kalpit1212
Manager
Manager
Joined: 25 Sep 2019
Posts: 57
Own Kudos [?]: 29 [0]
Given Kudos: 103
Send PM
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   08 Sep 2020, 13:37
can you please tell how did you factorize a cubic polynomial?
avatar
B
DippyDiva
Intern
Intern
Joined: 25 Dec 2019
Posts: 7
Own Kudos [?]: 0 [0]
Given Kudos: 9
Location: India
Schools: Booth '24 Darden '24 ESADE Foster '24 Fuqua '24 Haas '24 IESE '24 ISB '23 Judge '23 Kelley '24 Kellogg '24 LBS '24 NTU NUS Said '23 Bocconi Sloan '24 Tepper '24 Tuck '24 Yale '24
GMAT 1: 710 Q49 V34
GPA: 4
Send PM
Re: m is the smallest positive integer such that for any integer n ≥ m [#permalink] New post   08 Sep 2020, 20:19
Kalpit1212 wrote:
can you please tell how did you factorize a cubic polynomial?

You normally look for one number when substituted gives 0. In this case finding that one number was easy. So once you get '1' as a root n-1 becomes a factor. Divide the cubic by n-1 to get a quadratic that can easily be factorized.

Posted from my mobile device
User avatar
B
RastogiSarthak99
Manager
Manager
Joined: 20 Mar 2019
Posts: 147
Own Kudos [?]: 14 [0]
Given Kudos: 282
Location: India
Send PM
GMAT ToolKit User
Re: m is the smallest positive integer such that for any integer n m [#permalink] New post   26 Aug 2022, 09:29
This is solveable for when n = m and replace n for any of the values mentioned in the list.

Bunuel correct me if I am wrong but in GMAT options are listed in ascending or descending order of value right? This list here seems random. TIA.
User avatar
Chacotay
Intern
Intern
Joined: 25 Sep 2022
Posts: 4
Own Kudos [?]: 0 [0]
Given Kudos: 0
Send PM
Re: m is the smallest positive integer such that for any integer n m [#permalink] New post   26 Sep 2022, 03:00
One way to factorize polynomials is to use Rene Descartes Rational Root Theorem. In our case the factors of -5 are +1,-1,+5,-5 and the factors of the leading term 1 are +1,-1. Therefore, the rational possible roots of the polynomial are : +1,-1,+5,-5.
By checking it is obvious that +1 is a root of the polynomial and by a long division we find that
n^3 – 7n^2 + 11n - 5 = (n^2 – 6n + 5)(n – 1) = (n - 5)(n – 1)^2
However, (n – 1)^2 always positive. So, n – 5 > 0 => n > 5
Solution is n = 6 => D
GMAT Club Bot
gmatclubot
Re: m is the smallest positive integer such that for any integer n m [#permalink]
26 Sep 2022, 03:00
Moderators:
Bunuel
Math Expert
92918 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions