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

It is currently 20 Aug 2014, 03:12

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

For any integer k > 1, the term “length of an integer”

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
3 KUDOS received
Senior Manager
Senior Manager
avatar
Joined: 28 Aug 2010
Posts: 267
Followers: 3

Kudos [?]: 74 [3] , given: 11

GMAT Tests User
For any integer k > 1, the term “length of an integer” [#permalink] New post 22 Jan 2011, 11:31
3
This post received
KUDOS
4
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

45% (02:36) correct 55% (02:04) wrong based on 357 sessions
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

A. 5
B. 6
C. 15
D. 16
E. 18
[Reveal] Spoiler: OA

_________________

Verbal:new-to-the-verbal-forum-please-read-this-first-77546.html
Math: new-to-the-math-forum-please-read-this-first-77764.html
Gmat: everything-you-need-to-prepare-for-the-gmat-revised-77983.html
-------------------------------------------------------------------------------------------------
Ajit

Expert Post
24 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19027
Followers: 3359

Kudos [?]: 24348 [24] , given: 2677

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 22 Jan 2011, 11:49
24
This post received
KUDOS
Expert's post
2
This post was
BOOKMARKED
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.


Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.
_________________

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

Senior Manager
Senior Manager
avatar
Joined: 28 Aug 2010
Posts: 267
Followers: 3

Kudos [?]: 74 [0], given: 11

GMAT Tests User
Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 22 Jan 2011, 11:51
ah ! . awesome..thanks Bunuel !
_________________

Verbal:new-to-the-verbal-forum-please-read-this-first-77546.html
Math: new-to-the-math-forum-please-read-this-first-77764.html
Gmat: everything-you-need-to-prepare-for-the-gmat-revised-77983.html
-------------------------------------------------------------------------------------------------
Ajit

Expert Post
2 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19027
Followers: 3359

Kudos [?]: 24348 [2] , given: 2677

Re: length of primes [#permalink] New post 11 Feb 2012, 01:18
2
This post received
KUDOS
Expert's post
devinawilliam83 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

A 5
B 6
C 15
D 16
E 18


Merging similar topics. Ask if anything remains unclear.

Questions about the same concept:
length-of-an-integer-126368.html
for-any-positive-integer-n-the-length-of-n-is-defined-as-126740.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

Senior Manager
Senior Manager
avatar
Joined: 07 Sep 2010
Posts: 340
Followers: 3

Kudos [?]: 109 [0], given: 136

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 15 Mar 2012, 05:15
Bunuel wrote:

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.


Hi Bunuel,
I tried solving this question. However, I thought x and y to be different. That's why I put x= 2 and y = 3, in order to minimize the prime bases and thus maximize the powers of the primes.
Isnt the question implying x and y to be different. Otherwise the given equation x+3y<1000 is as good as x+3x<1000

..getting what I am trying to put across?
_________________

+1 Kudos me, Help me unlocking GMAT Club Tests

Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19027
Followers: 3359

Kudos [?]: 24348 [0], given: 2677

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 15 Mar 2012, 06:33
Expert's post
imhimanshu wrote:
Bunuel wrote:

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.


Hi Bunuel,
I tried solving this question. However, I thought x and y to be different. That's why I put x= 2 and y = 3, in order to minimize the prime bases and thus maximize the powers of the primes.
Isnt the question implying x and y to be different. Otherwise the given equation x+3y<1000 is as good as x+3x<1000

..getting what I am trying to put across?


We are not told that x and y are distinct, so we cannot assume this.
_________________

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

Manager
Manager
avatar
Joined: 17 Oct 2010
Posts: 81
Followers: 1

Kudos [?]: 70 [0], given: 26

GMAT Tests User
Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 15 May 2012, 09:27
Bunuel wrote:
imhimanshu wrote:
Bunuel wrote:

Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.


Hi Bunuel,
I tried solving this question. However, I thought x and y to be different. That's why I put x= 2 and y = 3, in order to minimize the prime bases and thus maximize the powers of the primes.
Isnt the question implying x and y to be different. Otherwise the given equation x+3y<1000 is as good as x+3x<1000

..getting what I am trying to put across?


We are not told that x and y are distinct, so we can not assume this. Next, even if we were told that they are distinct the answer still would be D: 2^8*3=768<100 also has the length of 8+1=9.



Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19027
Followers: 3359

Kudos [?]: 24348 [0], given: 2677

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 16 May 2012, 01:12
Expert's post
Joy111 wrote:
Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks


We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.
_________________

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

Manager
Manager
avatar
Joined: 17 Oct 2010
Posts: 81
Followers: 1

Kudos [?]: 70 [0], given: 26

GMAT Tests User
Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 16 May 2012, 01:19
Bunuel wrote:
Joy111 wrote:
Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks


We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.


Hi Bunuel

shouldn't we take 3*3^4= 3^5 ( adding the exponents with same base) and hence 9+5= 14 ?
Expert Post
1 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19027
Followers: 3359

Kudos [?]: 24348 [1] , given: 2677

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 16 May 2012, 01:22
1
This post received
KUDOS
Expert's post
Joy111 wrote:
Bunuel wrote:
Joy111 wrote:
Hi Bunuel

x+3y<1000 and if x and y are distinct then shouldn't this be
2^9+3^5<1000 for a length of 14 which is not among the options, so I guess the question didn't mean that they are distinct. Please can you explain once more if they are distinct how can the answer still be 16 (D)

Thanks


We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.


Hi Bunuel

shouldn't we take 3*3^4= 3^5 ( adding the exponents with same base) and hence 9+5= 14 ?


No. Please read the question carefully "what is the maximum possible sum of the length of x and the length of y". The length of x is 9 and the length of y is 4, so the sum is 9+4=13.
_________________

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

Manager
Manager
avatar
Joined: 17 Oct 2010
Posts: 81
Followers: 1

Kudos [?]: 70 [0], given: 26

GMAT Tests User
Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 16 May 2012, 03:18
We are not told that x and y are distinct. But if we were told that, the answer would be 13 not 14: x+3y=2^9+3*3^4=755<1,000 --> 9+4=13.[/quote]

Hi Bunuel

shouldn't we take 3*3^4= 3^5 ( adding the exponents with same base) and hence 9+5= 14 ?[/quote]

No. Please read the question carefully "what is the maximum possible sum of the length of x and the length of y". The length of x is 9 and the length of y is 4, so the sum is 9+4=13.[/quote]


Awesome , thanks ,really fell in the trap for that one !! + 1
2 KUDOS received
Manager
Manager
User avatar
Joined: 01 Nov 2010
Posts: 194
Location: India
Concentration: Technology, Marketing
GMAT Date: 08-27-2012
GPA: 3.8
WE: Marketing (Manufacturing)
Followers: 5

Kudos [?]: 19 [2] , given: 28

GMAT Tests User
Re: ps [#permalink] New post 09 Jun 2012, 21:30
2
This post received
KUDOS
here is my approach :
we know that : x > 1, y > 1, and x + 3y < 1000,
and it is given that length means no of factors.
for any value of x and y, the max no of factors can be obtained only if factor is smallest no & all factors are equal.
hence, lets start with smallest no 2.
2^1 =2
2^2 =4
2^3=8
2^4=16
2^5=32
2^6=64
2^7=128
2^8=256
2^9=512
2^10 =1024 (opps//it exceeds 1000, so, x can't be 2^10)
so, max value that X can take is 2^9 , for which has "length of integer" is 9.
now, since x =512 , & x+3y<1000
so, 3y<488
==> y<162
so, y can take any value which is less than 162. and to get the maximum no of factors of smallest integer, we can say y=2^7
for 2^7 has length of integer is 7.

SO, combined together : 9+7 = 16.
And is D.

Hope it will help.
_________________

kudos me if you like my post.

Attitude determine everything.
all the best and God bless you.

1 KUDOS received
Senior Manager
Senior Manager
User avatar
Joined: 13 Jan 2012
Posts: 304
Weight: 170lbs
GMAT 1: 730 Q48 V42
GMAT 2: 740 Q48 V42
WE: Analyst (Other)
Followers: 9

Kudos [?]: 64 [1] , given: 36

Re: ps [#permalink] New post 09 Jun 2012, 21:42
1
This post received
KUDOS
What's the source of this problem? Basically, you know that in order to get the longest "length", we'll want all 2's in the prime factorization. So how can we satisfy x + 3y < 1000 where x and y are both 2 to the nth power.

Let's start here:
2^8 + 3(2^8) = ?
256 + 768 = 1024

TOO HIGH, but that's interesting. It definitely looks like we can play around with this somehow to reach it. We know that it won't make sense to increase to 2^9 for y's value because 2^9 is 512 and 512*3 > 1000. How about the other way around?

2^9 + 3(2^7) = ?
512 + 3(128) = 896

9 + 7 = 16.

We know we won't be able to get much higher than that because 2^10 as x is the only other move we could try and that is > 1000 by itself. (It's 1024 which you should have memorized!) So answer is D = 16.

Hope that helps.
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19027
Followers: 3359

Kudos [?]: 24348 [0], given: 2677

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 12 Jun 2013, 04:25
Expert's post
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Number Properties: math-number-theory-88376.html

All DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
All PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59

_________________

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

Manager
Manager
avatar
Joined: 04 Nov 2012
Posts: 70
Schools: NTU '16 (A)
Followers: 0

Kudos [?]: 15 [0], given: 39

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 01 Aug 2013, 06:23
Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.


Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.



Hi Bunuel, Don't we need to check for the other case. i.e when try to maximise the length of Y rather than that of X??
Manager
Manager
avatar
Joined: 26 Jul 2012
Posts: 63
Followers: 0

Kudos [?]: 4 [0], given: 8

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 29 Aug 2013, 19:37
12bhang wrote:
Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.


Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.



Hi Bunuel, Don't we need to check for the other case. i.e when try to maximise the length of Y rather than that of X??



I think we can try both cases to see which gives you the maximum Length.

1) maximize Y

x + 3 y < 1000
2^9 = 512
2 ^8 = 256
so y = 2^ 8
and now x is 999 - 3(256) = 231
x can be 2^7 = 128
So total length = 8 + 7 = 15

2) maximize x

x + 3 y < 1000
2^9 = 512
2 ^ 10 = 1024
length of x = 9
now, y = (999 - 512 ) / 3 = 162.x
or y is 2^7

Total length is 9+7 = 16.

Now, you might want to NOT do the maximization of y because you know that most 2s will be in x and NOT y. For example, we have to first realize that we want the number with the MOST 2s in it. x can be that number as illustrated by this example: If y = 512 -> 512 * 3 > 1000 and if x = 512 , well x can be 512. So, we would get a length of 9 out of it.

If we were doing, x + y < 1000 then the lengths can be inter-changed; however, because of the 3 next to y, we know that the length of y will have to be LESS than the length of x.

Answer D.
Senior Manager
Senior Manager
avatar
Joined: 07 Apr 2012
Posts: 341
Followers: 0

Kudos [?]: 18 [0], given: 47

Re: For any integer k > 1, the term “length of an integer” [#permalink] New post 11 Nov 2013, 09:17
Bunuel wrote:
ajit257 wrote:
For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

a. 5
b. 6
c. 15
d. 16
e. 18

Can some explain an elegant way of doing such a problem which would take less time.


Basically the length of an integer is the sum of the powers of its prime factors. For example the length of 24 is 4 because 24=2^3*3^1 --> 3+1=4.

Given: x+3y<1,000. Now, to maximize the length of x or y (to maximize the sum of the powers of their primes) we should minimize their prime bases. Minimum prime base is 2: so if x=2^9=512 then its length is 9 --> 512+3y<1,000 --> y<162.7 --> maximum length of y can be 7 as 2^7=128 --> 9+7=16.

Answer: D.

Hi Bunuel,

How did you know to start with "x" and not "y"?
What is the logic?
When I start with "y" I get a different result....
Re: For any integer k > 1, the term “length of an integer”   [#permalink] 11 Nov 2013, 09:17
    Similar topics Author Replies Last post
Similar
Topics:
2 Experts publish their posts in the topic For any positive integer n, n>1, the "length" of n is the Splendidgirl666 3 21 Jan 2012, 06:25
Source:MGMAT CAT For any integer k > 1, the term length LiveStronger 6 02 Nov 2008, 07:29
For any integer k > 1, the term length of an integer dancinggeometry 1 16 Sep 2008, 07:43
For any integer k > 1, the term length of an integer applecrisp 2 08 Dec 2007, 10:48
1 Experts publish their posts in the topic For any positive integer n, n>1, the "length" of n is the Sumithra Sen 3 18 Aug 2007, 11:24
Display posts from previous: Sort by

For any integer k > 1, the term “length of an integer”

  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®.