Find all School-related info fast with the new School-Specific MBA Forum

It is currently 24 Jul 2014, 15:31

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

For any positive integer n, the length of n is defined as

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
1 KUDOS received
Director
Director
avatar
Status: Preparing for the 4th time -:(
Joined: 25 Jun 2011
Posts: 563
Location: United Kingdom
Concentration: International Business, Strategy
GMAT Date: 06-22-2012
GPA: 2.9
WE: Information Technology (Consulting)
Followers: 12

Kudos [?]: 265 [1] , given: 217

GMAT Tests User
For any positive integer n, the length of n is defined as [#permalink] New post 29 Jan 2012, 16:15
1
This post received
KUDOS
4
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

65% (01:47) correct 35% (01:02) wrong based on 212 sessions
For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.
[Reveal] Spoiler: OA

_________________

Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610 :-(

Expert Post
9 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 18729
Followers: 3239

Kudos [?]: 22351 [9] , given: 2615

Re: 2 digit positive integers with length 6 [#permalink] New post 29 Jan 2012, 16:21
9
This post received
KUDOS
Expert's post
2
This post was
BOOKMARKED
enigma123 wrote:
For any positive integer n, the length of n is defined as number of prime factors whose product is n, For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?

A. 0
B. 1
C. 2
D. 3
E. 4

I need to understand the concept behind solving this question please.


Basically the length of the integer is the sum of the powers of its prime factors.

Length of six means that the sum of the powers of primes of the two-digit integer must be 6. First we can conclude that 5 can not be a factor of this integer as the smallest integer with the length of six that has 5 as prime factor is 2^5*5=160 (length=5+1=6), not a two-digit integer.

The above means that the primes of the two-digit integers we are looking for can be only 2 and/or 3. n=2^p*3^q, p+q=6.

Let's start with the highest value of p:
n=2^6*3^0=64 (length=6+0=6);
n=2^5*3^1=96 (length=5+1=6);

n=2^4*3^2=144 (length=4+2=6) not good as 144 is a three digit integer.

Answer: C.

Questions about the same concept to practice:
length-of-an-integer-126368.html
for-any-integer-k-1-the-term-length-of-an-integer-108124.html

Hope it helps.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Director
Director
avatar
Status: Preparing for the 4th time -:(
Joined: 25 Jun 2011
Posts: 563
Location: United Kingdom
Concentration: International Business, Strategy
GMAT Date: 06-22-2012
GPA: 2.9
WE: Information Technology (Consulting)
Followers: 12

Kudos [?]: 265 [0], given: 217

GMAT Tests User
Re: For any positive integer n, the length of n is defined as [#permalink] New post 29 Jan 2012, 16:46
thanks Bunuel for a very thorough explanation.
_________________

Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610 :-(

Intern
Intern
avatar
Joined: 26 Jan 2012
Posts: 7
Location: United States
Concentration: General Management, Marketing
GMAT 1: 690 Q48 V37
WE: Design (Retail)
Followers: 0

Kudos [?]: 5 [0], given: 4

Re: For any positive integer n, the length of n is defined as [#permalink] New post 30 Jan 2012, 08:01
Answer is c! 2x2x2x2x2x2 & 2x2x2x2x2x3
1 KUDOS received
Senior Manager
Senior Manager
User avatar
Joined: 13 Aug 2012
Posts: 464
Concentration: Marketing, Finance
GMAT 1: Q V0
GPA: 3.23
Followers: 14

Kudos [?]: 179 [1] , given: 11

GMAT ToolKit User GMAT Tests User
For any positive integer n, the length of n is defined as number [#permalink] New post 12 Dec 2012, 20:32
1
This post received
KUDOS
My Solution:

Try increasing prime numbers with length 6:
Trial 1: 2^6=64 Valid
Trial 2: 3^6 =729 Invalid

This means our candidate 2-digit numbers have combinations of 2 and 3

2^6=64
2^5x3^1=96
2^4x3^2=144 Invalid

Answer: Two
_________________

Impossible is nothing to God.

Senior Manager
Senior Manager
avatar
Joined: 16 Feb 2012
Posts: 259
Concentration: Finance, Economics
Followers: 4

Kudos [?]: 45 [0], given: 102

GMAT ToolKit User
For any positive integer n, the length of n is defined as number [#permalink] New post 10 Feb 2013, 02:35
2x2x2x2x2x2 = 2^6 = 64
2x2x2x2x2x3 = 2^5x3 = 96

These are only two possible solutions, therefore the answer is C.
_________________

Kudos if you like the post!

Failing to plan is planning to fail.

Intern
Intern
avatar
Joined: 30 Apr 2010
Posts: 21
Followers: 0

Kudos [?]: 7 [0], given: 2

Re: For any positive integer n, the length of n is defined as [#permalink] New post 08 Mar 2013, 15:13
Try the smallest possible value first: In this case it is 2^6 which equals 64.

If we replace the last 2 with 3, then we have 2^5*3 = 96

From here we can positively assume that any other number will have more than 2 digits. So the answer is (C) 2 numbers that have length 6 and are only 2 digits.
SVP
SVP
User avatar
Joined: 09 Sep 2013
Posts: 1725
Followers: 165

Kudos [?]: 33 [0], given: 0

Premium Member
Re: For any positive integer n, the length of n is defined as [#permalink] New post 09 Mar 2014, 00:18
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

Director
Director
avatar
Joined: 03 Aug 2012
Posts: 883
Concentration: General Management, General Management
Schools: IIM A '15
GMAT 1: 630 Q47 V29
GMAT 2: 680 Q50 V32
GPA: 3.7
Followers: 11

Kudos [?]: 161 [0], given: 299

Premium Member CAT Tests
Re: For any positive integer n, the length of n is defined as [#permalink] New post 19 Apr 2014, 23:45
Length of a number = Total number of prime factors of the number.

Any composite number can be represented as a product of prime numbers as shown below:

N= 2^x * 3^y * 5^z * 7^a ...so on

Since our requirement is a two digit number, we shall raise maximum power for smallest prime factor.

N= 2^6 = 64 has Length 6

N= 2^5 * 3 = 96 has length 6

N=2^4 * 3^2 = 144 is a three digit number and so on the other combinations would reveal 3 digit numbers.

Hence there are only 2 numbers
_________________

Rgds,
TGC!
_____________________________________________________________________
I Assisted You => KUDOS Please
_____________________________________________________________________________

Intern
Intern
User avatar
Joined: 27 Sep 2013
Posts: 17
Location: Netherlands
Followers: 0

Kudos [?]: 5 [0], given: 0

Re: For any positive integer n, the length of n is defined as [#permalink] New post 30 Apr 2014, 23:48
Just writing it out took me .46 sec.:

length of 6, lets take the lowest prime factor 2

2x2x2x2x2x2 = 64

Now substitute the last 2 by a 3, and see the solution gets 96. We can think of what will happen when we substitute another 2 for a three.

Hence, C (2)
Re: For any positive integer n, the length of n is defined as   [#permalink] 30 Apr 2014, 23:48
    Similar topics Author Replies Last post
Similar
Topics:
2 For any positive integer n, the length of n is defined as jimjohn 3 26 Dec 2007, 19:16
For any positive integer n, the length of n is defined as gmatnub 4 23 Dec 2007, 01:19
For any positive integer n, the length of n is defined as r019h 1 23 Oct 2007, 08:43
For any positive integer n, the length of n is defined as zakk 1 08 Sep 2007, 12:59
For any positive integer n, the length of n is defined as khush 3 03 Apr 2006, 13:25
Display posts from previous: Sort by

For any positive integer n, the length of n is defined as

  Question banks Downloads My Bookmarks Reviews Important topics  


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

Powered by phpBB © phpBB Group and phpBB SEO

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®.