|
Author |
Message |
|
TAGS:
|
|
|
Manager
Joined: 19 Nov 2007
Posts: 228
Followers: 1
Kudos [?]:
55
[0], given: 1
|
For how many values of k is 12^12 the least common multiple [#permalink]
 12 Nov 2009, 04:38
Question Stats:
28% (01:02) correct
71% (08:45) wrong based on 7 sessions
For how many values of k is 12^12 the least common multiple of the positive integers 6^6, 8^8 and k?
A. 23
B. 24
C. 25
D. 26
E. 27
Last edited by Bunuel on 30 Mar 2012, 01:30, edited 1 time in total.
Edited the question and added the OA
|
|
|
|
|
|
|
Manager
Joined: 29 Oct 2009
Posts: 212
Followers: 49
Kudos [?]:
419
[3] , given: 18
|
Re: least common multiple -12^12 [#permalink]
12 Nov 2009, 05:12
3
This post received
KUDOS
jade3 wrote: For how many values of k is 12^12 the least common multiple of the positive integers 6^6, 8^8 and k?
A. 23
B. 24
C. 25
D. 26
E. 27 6^6 = (2^6)*(3^6)8^8 = 2^{24}Now we know that the least common multiple of the above two numbers and k is: 12^{12} = (2*2*3)^{12} = (2^{24})*(3^{12})Thus, k will also be in the form of : (2^a)*(3^b)Now, b has to be equal to 12 since in order for (2^{24})*(3^{12}) to be a common multiple, at least one of the numbers must have the terms 2^{24} and 3^{12} as its factors. (not necessarily the same number). We can see that 8^8 already takes care of the 2^{24} part. Thus, k has to take care of the 3^{12} part of the LCM. This means that the value k is (2^a)*(3^{12}) where a can be any value from 0 to 24 (both inclusive) without changing the value of the LCM. Thus K can have 25 values. Choice (c). Cheers.
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942
Word Problems Made Easy!
1) Translating the English to Math : word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : distance-speed-time-word-problems-made-easy-87481.html
|
|
|
|
|
|
Manager
Joined: 29 Oct 2009
Posts: 212
Followers: 49
Kudos [?]:
419
[0], given: 18
|
Re: least common multiple -12^12 [#permalink]
12 Nov 2009, 05:15
Quote: 8^8 = 2^24
8^8 = 2^(24) Similarly for the other numbers. Sorry for that confusion. Wasn't able to get a 2 digit power using the math function. If any one knows how to do it please do let me know. Cheers.
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942
Word Problems Made Easy!
1) Translating the English to Math : word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : distance-speed-time-word-problems-made-easy-87481.html
|
|
|
|
|
|
GMAT Club team member
Joined: 02 Sep 2009
Posts: 12116
Followers: 1879
Kudos [?]:
10132
[0], given: 965
|
Re: least common multiple -12^12 [#permalink]
12 Nov 2009, 05:42
|
|
|
|
|
|
Manager
Joined: 29 Oct 2009
Posts: 212
Followers: 49
Kudos [?]:
419
[0], given: 18
|
Re: least common multiple -12^12 [#permalink]
12 Nov 2009, 05:53
Bunuel wrote: Edited your post. Please check if I didn't mess it up accidentally. To get two digit power just put the power in {}, eg. 2^{24} and mark with m button. Nope, you didn't mess it up.. Only made it better! Thanks Brunel! Infact thanks^{10} !! 
_________________
Click below to check out some great tips and tricks to help you deal with problems on Remainders!
compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942
Word Problems Made Easy!
1) Translating the English to Math : word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : distance-speed-time-word-problems-made-easy-87481.html
|
|
|
|
|
|
Manager
Joined: 19 Nov 2007
Posts: 228
Followers: 1
Kudos [?]:
55
[0], given: 1
|
Re: least common multiple -12^12 [#permalink]
12 Nov 2009, 09:43
sriharimurthy wrote:
6^6 = (2^6)*(3^6)
8^8 = 2^{24}
Now we know that the least common multiple of the above two numbers and k is:
12^{12} = (2*2*3)^{12} = (2^{24})*(3^{12})
Thus, k will also be in the form of : (2^a)*(3^b)
Now, b has to be equal to 12 since in order for (2^{24})*(3^{12}) to be a common multiple, at least one of the numbers must have the terms 2^{24} and 3^{12} as its factors. (not necessarily the same number).
We can see that 8^8 already takes care of the 2^{24} part.
Thus, k has to take care of the 3^{12} part of the LCM.
This means that the value k is (2^a)*(3^{12}) where a can be any value from 0 to 24 (both inclusive) without changing the value of the LCM.
Thus K can have 25 values. Choice (c).
Cheers.
You are spot on The answer is indeed C
|
|
|
|
|
|
Intern
Joined: 22 Jan 2012
Posts: 26
Followers: 0
Kudos [?]:
5
[0], given: 11
|
Re: least common multiple -12^12 [#permalink]
29 Mar 2012, 14:54
sriharimurthy wrote: jade3 wrote: Thus, k will also be in the form of : (2^a)*(3^b)
Hi, I'm trying to understand this question.. the explanation seems good, but I still can't seem to get a grasp of it.. why can we say that K is also in the form of (2^a)*(3^b) ?? also, how do we consider the 6^6 term in this explanation? any help is appreciated, Thanks!
|
|
|
|
|
|
GMAT Club team member
Joined: 02 Sep 2009
Posts: 12116
Followers: 1879
Kudos [?]:
10132
[1] , given: 965
|
Re: least common multiple -12^12 [#permalink]
30 Mar 2012, 01:29
1
This post received
KUDOS
essarr wrote: sriharimurthy wrote: jade3 wrote: Thus, k will also be in the form of : (2^a)*(3^b)
Hi, I'm trying to understand this question.. the explanation seems good, but I still can't seem to get a grasp of it.. why can we say that K is also in the form of (2^a)*(3^b) ?? also, how do we consider the 6^6 term in this explanation? any help is appreciated, Thanks! A. 23 B. 24 C. 25 D. 26 E. 27 We are given that 12^{12}=2^{24}*3^{12} is the least common multiple of the following three numbers: 6^6=2^6*3^6; 8^8 = 2^{24}; and k; First notice that k cannot have any other primes other than 2 or/and 3, because LCM contains only these primes. Now, since the power of 3 in LCM is higher than the powers of 3 in either the first number or in the second, than k must have 3^{12} as its multiple (else how 3^{12} would appear in LCM?). Next, k can have 2 as its prime in ANY power ranging from 0 to 24, inclusive (it cannot have higher power of 2 since LCM limits the power of 2 to 24). For example k can be: 2^0*3^{12}=3^{12}; 2^1*3^{12}; 2^2*3^{12}; ... 2^{24}*3^{12}=12^{12}=LCM. So, k can take total of 25 values. Answer: C. 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
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. NEW!!! ,11 Mixed Questions NEW!!!, 12 Fresh Meat NEW!!!
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. NEW!!!, 11 New DS set. NEW!!!
What are GMAT Club Tests?
25 extra-hard Quant Tests
Find out what's new at GMAT Club - latest features and updates
|
|
|
|
|
|
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 3170
Location: Pune, India
Followers: 598
Kudos [?]:
2129
[2] , given: 97
|
Re: least common multiple -12^12 [#permalink]
30 Mar 2012, 04:43
2
This post received
KUDOS
essarr wrote: sriharimurthy wrote: jade3 wrote: Thus, k will also be in the form of : (2^a)*(3^b)
Hi, I'm trying to understand this question.. the explanation seems good, but I still can't seem to get a grasp of it.. why can we say that K is also in the form of (2^a)*(3^b) ?? also, how do we consider the 6^6 term in this explanation? any help is appreciated, Thanks! Here is my explanation: LCM (Least Common Multiple) of 3 numbers a, b and c would be a multiple of each of these 3 numbers. So for every prime factor in these numbers, LCM would have the highest power available in any number e.g. a = 2*5b = 2*5*7^2c = 2^4*5^2What is the LCM of these 3 numbers? It is 2^4*5^2*7^2 Every prime factor will be included and the power of every prime factor will be the highest available in any number. So if, a = 2^6*3^6b = 2^{24}k = ? LCM = 2^{24}*3^{12} What values can k take? First of all, LCM has 3^{12}. From where did it get 3^{12}? a and b have a maximum 3^6. This means k must have 3^{12}. Also, LCM has 2^{24} which is available in b. So k needn't have 2^{24}. It can have 2 to any power as long as it is less than or equal to 24. k can be 2^{0}*3^{12} or 2^{1}*3^{12} or 2^{2}*3^{12} ... 2^{24}*3^{12}The power of 2 in k cannot exceed 24 because then, the LCM would have the higher power. What about some other prime factor? Can k be 2^{4}*3^{12}*5? No, because then the LCM would have 5 too. So k can take 25 values only
_________________
Karishma
Veritas Prep | GMAT Instructor
My Blog
Save 10% on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.
Veritas Prep Reviews
|
|
|
|
|
|
Intern
Joined: 22 Jan 2012
Posts: 26
Followers: 0
Kudos [?]:
5
[0], given: 11
|
Re: For how many values of k is 12^12 the least common multiple [#permalink]
31 Mar 2012, 12:48
ahhhhh I see it now; thanks so much bunuel & karishma, that clarified it
|
|
|
|
|
|
|
Re: For how many values of k is 12^12 the least common multiple
[#permalink]
31 Mar 2012, 12:48
|
|
|
|
|
|
|
|
|
|
|
close
