It is currently 20 Oct 2017, 13:12

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.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

The function f(m) is defined for all positive integers m as

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

3 KUDOS received
Intern
Intern
avatar
Joined: 02 Nov 2010
Posts: 8

Kudos [?]: 55 [3], given: 1

The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 26 Jan 2011, 14:00
3
This post received
KUDOS
30
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

66% (01:06) correct 34% (01:19) wrong based on 729 sessions

HideShow timer Statistics

The function f(m) is defined for all positive integers m as the product of m + 4, m + 5, and m + 6. If n is a positive integer, then f(n) must be divisible by which one of the following numbers?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 11
[Reveal] Spoiler: OA

Kudos [?]: 55 [3], given: 1

Expert Post
22 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 41892

Kudos [?]: 129017 [22], given: 12187

Re: Defined Functions [#permalink]

Show Tags

New post 26 Jan 2011, 14:38
22
This post received
KUDOS
Expert's post
20
This post was
BOOKMARKED
PASSINGGMAT wrote:
The function f(m) is defined for all positive integers m as the product of m + 4, m + 5, and m + 6. If n is a positive integer, then f(n) must be divisible by which one of the following numbers?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 11

Can you explain how to answer this please. Thank you.


Given: \(f(n)=(n+4)(n+5)(n+6)\), where \(n\) is a positive integer. Question: \(f(n)\) must be divisible by which one of the following numbers.

Now, \((n+4)(n+5)(n+6)\) is the product of 3 consecutive integers so out of them one is definitely divisible by 3 and at least one is divisible by 2, so \(f(n)\) must be divisible by 2*3=6.

Answer: C.

Generally out of ANY \(k\) consecutive integers one is always divisible by \(k\) and at least one by \(k-1\), \(k-2\), ... For example out of ANY 5 consecutive integers there is one which is divisible by 5, and at least one which is divisible by 4, 3, and 2. That's because an integer divided by an integer \(k\) can give a remainder of: 0 (when it's divisible by \(k\)), 1, 2, ..., or \(k-1\) (total of \(k\) different remainders from 0 to \(k-1\)), so out of \(k\) consecutive integers there definitely will be one which gives a reminder of zero, so divisible by \(k\).

Which give us the following property: the product of \(k\) consecutive integers is always divisible by \(k!\), so by \(k\) too. For example: given \(k=4\) consecutive integers \(\{3,4,5,6\}\) --> the product of 3*4*5*6 is 360, which is divisible by 4!=24.

If we apply this property to the original question we'll have that the product of given 3 consecutive integers \((n+4)(n+5)(n+6)\) must be divisible by 3!=6.

Hope it's clear.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 129017 [22], given: 12187

1 KUDOS received
Director
Director
avatar
Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Joined: 03 Feb 2011
Posts: 871

Kudos [?]: 396 [1], given: 123

Reviews Badge
Re: Defined Functions [#permalink]

Show Tags

New post 08 Mar 2011, 20:14
1
This post received
KUDOS
Product of Any three consecutive integers must be divisible by 6.

Kudos [?]: 396 [1], given: 123

Expert Post
1 KUDOS received
EMPOWERgmat Instructor
User avatar
P
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 9984

Kudos [?]: 3412 [1], given: 172

Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: 340 Q170 V170
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 29 Mar 2015, 19:21
1
This post received
KUDOS
Expert's post
Hi All,

This question is based on a couple of Number Properties; as such, you can also TEST VALUES to get to the solution.

We're told that the f(M) = (M+4)(M+5)(M+6) for all POSITIVE integers. We're asked which of the following numbers MUST divide into f(N).

IF....
N = 1
f(1) = (5)(6)(7)

At this point, you can either multiply out the numbers and check the 5 answer choices against that product OR prime factor the f(1)....

(5)(6)(7) = (5)(2)(3)(7)

Of the 5 answer choices, only 2 of them divide into this product (Answers B and C; 5 and 6).

From here, we should look to try to eliminate one of the options. It's actually not that hard....

IF....
N = 2
f(2) = (6)(7)(8)

Looking at this, we can see that 6 IS a factor while 5 is NOT.

Final Answer:
[Reveal] Spoiler:
C


GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free
  Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Kudos [?]: 3412 [1], given: 172

Manager
Manager
User avatar
Joined: 07 Jan 2010
Posts: 143

Kudos [?]: 95 [0], given: 57

Location: So. CA
WE 1: 2 IT
WE 2: 4 Software Analyst
Re: Defined Functions [#permalink]

Show Tags

New post 27 Jan 2011, 22:45
awesome explanation

Kudos [?]: 95 [0], given: 57

Senior Manager
Senior Manager
User avatar
Joined: 08 Nov 2010
Posts: 394

Kudos [?]: 128 [0], given: 161

WE 1: Business Development
GMAT ToolKit User
Re: Defined Functions [#permalink]

Show Tags

New post 29 Jan 2011, 12:59
Ye, Bunuel - u rocked it. +1.
_________________

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 128 [0], given: 161

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 16635

Kudos [?]: 273 [0], given: 0

Premium Member
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 03 Oct 2013, 19:46
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 273 [0], given: 0

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 16635

Kudos [?]: 273 [0], given: 0

Premium Member
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 14 Nov 2014, 00:34
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 273 [0], given: 0

Intern
Intern
User avatar
Joined: 22 Nov 2012
Posts: 22

Kudos [?]: 7 [0], given: 90

Location: United States
GMAT ToolKit User
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 29 Mar 2015, 11:19
Bunuel

Very informative explanation Bunuel!

Thanks :)!
_________________

GMAT,
It is not finished untill I win!!!

Kudos [?]: 7 [0], given: 90

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 16635

Kudos [?]: 273 [0], given: 0

Premium Member
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 14 Apr 2016, 23:24
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 273 [0], given: 0

Intern
Intern
avatar
Joined: 31 May 2015
Posts: 1

Kudos [?]: [0], given: 40

Concentration: Marketing, Finance
Schools: Wharton '17
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 30 Sep 2016, 22:33
[align=]I tried the sum with f(13), which is equal to 17 X 18 X 19 (this number isn't divisible by 6)?[/align]
_________________

PNH0505

Kudos [?]: [0], given: 40

Expert Post
EMPOWERgmat Instructor
User avatar
P
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 9984

Kudos [?]: 3412 [0], given: 172

Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: 340 Q170 V170
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 01 Oct 2016, 16:45
Hi pnh0505,

You are correct that when you use M=13, then f(13) = (17)(18)(19). However, that product IS divisible by 6 - because 18 is divisible by 6.

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free
  Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Kudos [?]: 3412 [0], given: 172

Senior Manager
Senior Manager
avatar
B
Joined: 05 Jan 2017
Posts: 435

Kudos [?]: 59 [0], given: 15

Location: India
Premium Member
Re: The function f(m) is defined for all positive integers m as [#permalink]

Show Tags

New post 01 Mar 2017, 01:24
In three consecutive integers we can say that one of it will be an even number and one will a multiple of three.
To the product will be definitely the product of 6.
Option C

Kudos [?]: 59 [0], given: 15

Re: The function f(m) is defined for all positive integers m as   [#permalink] 01 Mar 2017, 01:24
Display posts from previous: Sort by

The function f(m) is defined for all positive integers m as

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.