For how many values of k is 12^12 the least common multiple : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 19 Jan 2017, 17:44

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# For how many values of k is 12^12 the least common multiple

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

### Hide Tags

Manager
Joined: 19 Nov 2007
Posts: 225
Followers: 1

Kudos [?]: 255 [0], given: 1

For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

12 Nov 2009, 03:38
25
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

45% (02:15) correct 55% (02:51) wrong based on 175 sessions

### HideShow timer Statistics

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
[Reveal] Spoiler: OA

Last edited by Bunuel on 30 Jun 2016, 07:51, edited 2 times in total.
Edited the question and added the OA
Manager
Joined: 29 Oct 2009
Posts: 211
GMAT 1: 750 Q50 V42
Followers: 103

Kudos [?]: 1281 [6] , given: 18

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

12 Nov 2009, 04:12
6
This post received
KUDOS
3
This post was
BOOKMARKED
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!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html

Manager
Joined: 29 Oct 2009
Posts: 211
GMAT 1: 750 Q50 V42
Followers: 103

Kudos [?]: 1281 [0], given: 18

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

12 Nov 2009, 04:15
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!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html

Math Expert
Joined: 02 Sep 2009
Posts: 36567
Followers: 7081

Kudos [?]: 93205 [0], given: 10553

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

12 Nov 2009, 04:42
sriharimurthy wrote:
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.

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.
_________________
Manager
Joined: 29 Oct 2009
Posts: 211
GMAT 1: 750 Q50 V42
Followers: 103

Kudos [?]: 1281 [0], given: 18

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

12 Nov 2009, 04:53
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!
http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy!
1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html
3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html

Manager
Joined: 19 Nov 2007
Posts: 225
Followers: 1

Kudos [?]: 255 [0], given: 1

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

12 Nov 2009, 08:43
1
This post was
BOOKMARKED
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: 22
Followers: 0

Kudos [?]: 36 [0], given: 11

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

29 Mar 2012, 13:54
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!
Math Expert
Joined: 02 Sep 2009
Posts: 36567
Followers: 7081

Kudos [?]: 93205 [2] , given: 10553

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

30 Mar 2012, 00:29
2
This post received
KUDOS
Expert's post
6
This post was
BOOKMARKED
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!

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

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.
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7125
Location: Pune, India
Followers: 2135

Kudos [?]: 13655 [6] , given: 222

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

30 Mar 2012, 03:43
6
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
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.
$$a = 2*5$$
$$b = 2*5*7^2$$
$$c = 2^4*5^2$$

What 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,
$$a = 2^6*3^6$$
$$b = 2^{24}$$
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

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 22 Jan 2012 Posts: 22 Followers: 0 Kudos [?]: 36 [0], given: 11 Re: For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 31 Mar 2012, 11:48 ahhhhh I see it now; thanks so much bunuel & karishma, that clarified it Senior Manager Joined: 15 Aug 2013 Posts: 328 Followers: 0 Kudos [?]: 53 [0], given: 23 Re: For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 23 Aug 2014, 09:21 Bunuel wrote: essarr 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 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. Hi Bunuel, I can see why K needs to have 3^12, but can't K have other values with the base 2? Meaning, why does the range only go from 2^0 to 2^24, why can't it be 2^-5 etc? GMAT Club Legend Joined: 09 Sep 2013 Posts: 13458 Followers: 575 Kudos [?]: 163 [0], given: 0 Re: For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 15 Sep 2015, 22:54 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. _________________ Intern Joined: 16 Nov 2015 Posts: 17 Followers: 0 Kudos [?]: 0 [0], given: 64 Re: For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 08 Dec 2015, 05:16 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 There are 3 numbers: 6^6 (in prime factors that is 2^6 * 3^6), 8^8 (that is 2^24) and k. LCM of these three numbers is given as: 12^12 (that is 3^12 * 2^24 ) First we can ignore k and find the LCM of the given two numbers (2^6 * 3^6) and (2^24) That is => 3^6 * 2^24 (Note that LCM of any two -or more- numbers is the product of all distinct prime factors with the greatest powers.) So if 3^6 * 2^24 (LCM of the given two numbers) and k has a LCM of 3^12 * 2^24 then k must have the factor 3^12 (this is a necessity because other number is limited with 2^6 ) On the other hand -besides 3^12- k can take prime 2 to the power of 0 to 24 (2^0 to 2^24) Therefore k can be any of the following: (3^12 and 2^0) or (3^12 and 2^1) or (3^12 and 2^2), ....., (3^12 and 2^24) that is 25 in total. (I think this is a 700 level question) BSchool Forum Moderator Joined: 12 Aug 2015 Posts: 1897 Followers: 49 Kudos [?]: 363 [0], given: 453 Re: For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 14 Mar 2016, 00:31 Great Question Here values of k are => 3^12 * 2^p for p=> [0,24] so 25 values hence C _________________ Mock Test -1 (Integer Properties Basic Quiz) ---> http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1676182 Mock Test -2 (Integer Properties Advanced Quiz) --->http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1765951 Give me a hell yeah ...!!!!! Director Joined: 04 Jun 2016 Posts: 656 GMAT 1: 750 Q49 V43 Followers: 50 Kudos [?]: 199 [0], given: 36 For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 30 Jun 2016, 10:09 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 Lets use a quick example What is the LCM of 2,4,9,12, Factorise all the numbers one by one and write them in prime numbers raised to exponent form $$(Prime1)^m$$ X $$(Prime2)^m$$... $$2=2^1$$ $$4=2*2==>2^2$$ $$9=3*3==>3^2$$ $$12=4*3==>2*2*3==>2^2 * 3^1$$ NOW LCM OF THESE NUMBERS WILL TAKE THE HIGHEST POWER OF EACH PRIME FROM EACH NUMBER (ONE TIMES ONLY) So LCM = $$2^2 * 3^3$$==> 4*9=36 Notice how $$2^1$$ and $$3^1$$ are not contributing towards the LCM at all. Now apply the same logic to your question You already know LCM is = $$12^{12}=(4*3)^{12}$$==> $$(2^2*3)^{12}$$ ==> $$2^{24}*3^{12}$$ Similarly $$6^6= (2*3)^6==>2^6*3^6$$ So we know $$6^6$$ is neither contributing 2's or 3's towards the LCM $$8^8= (2^3)^8==> 2^{24}$$ , So we know 8 is contributing all the $$2^{24}$$towards our LCM Now we need a $$3^{12}$$ to reach the LCM Since K is the only remaining digit therefore K must contribute $$3^{12}$$ but it is also possible K can or cannot have $$2^m$$ in it also and the values of $$2^m$$ can vary from $$2^0 to 2^{24}$$ Remember for LCM we take the highest power, so $$2^{24}$$ can be common in $$8^8$$ as well as K therefore total values of 2 in k = $$2^0 to 2^{24}$$ (Total=25) and one compulsory value of $$3^{12}$$ (total= 1) Total=26 values Answer is D Why am in overshooting by 1? _________________ Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly. FINAL GOODBYE :- 17th SEPTEMBER 2016. Last edited by LogicGuru1 on 01 Jul 2016, 00:38, edited 1 time in total. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2135 Kudos [?]: 13655 [0], given: 222 Re: For how many values of k is 12^12 the least common multiple [#permalink] ### Show Tags 30 Jun 2016, 23:39 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 Quote: (Total=25) and one compulsory value of 3^12 Total=26 values Answer is D Here is the problem in your solution. When you say the possible values vary from 2^0 to 2^24 (that is 25 values) AND another value is 3^12, you are double counting 3^12. Note that 2^0 = 1. So 2^0*3^12 = 3^12 Hence you have only 25 values. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Director
Joined: 04 Jun 2016
Posts: 656
GMAT 1: 750 Q49 V43
Followers: 50

Kudos [?]: 199 [0], given: 36

For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

01 Jul 2016, 00:50
VeritasPrepKarishma wrote:
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

Quote:

(Total=25) and one compulsory value of 3^12

Total=26 values
Answer is D

Here is the problem in your solution.
When you say the possible values vary from 2^0 to 2^24 (that is 25 values) AND another value is 3^12, you are double counting 3^12.
Note that 2^0 = 1. So
2^0*3^12 = 3^12

Hence you have only 25 values.

Thanks Karishma ,
Just to clarify one more doubt,
4= 2*2 = $$2^2$$ ==> The genreal form is $$2^q$$
Total possible factors of 4 = q+1 = 2+1 = 3 {1,2,4}
IS this the same thing that you mentioned :- I am counting 1 in $$3^{12}$$ and also in $$2^0$$ and i need to drop it one time.? RIGHT ??

In all such questions, does one need to ignore "1" in the final count ?
_________________

Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly.
FINAL GOODBYE :- 17th SEPTEMBER 2016.

Senior Manager
Status: Exam scheduled!!
Joined: 05 Sep 2016
Posts: 411
Location: United States (WI)
Concentration: Marketing, Technology
WE: Other (Law)
Followers: 3

Kudos [?]: 13 [0], given: 241

Re: For how many values of k is 12^12 the least common multiple [#permalink]

### Show Tags

29 Nov 2016, 17:30
K can take on any of the following values:

(3^12)
(3^12)(2)
(3^12)(2^2)
(3^12)(2^3)
(3^12)(2^4)
(3^12)(2^5)
(3^12)(2^6)
(3^12)(2^7)
(3^12)(2^8)
(3^12)(2^9)
(3^12)(2^10)
(3^12)(2^11)
(3^12)(2^12)
(3^12)(2^13)
.
.
.
(3^12)(2^24)

Thus, there are 25 values that K can take on.

C.
Re: For how many values of k is 12^12 the least common multiple   [#permalink] 29 Nov 2016, 17:30
Similar topics Replies Last post
Similar
Topics:
Which of the following is the least common multiple of 136 and 204? 3 01 Dec 2016, 00:59
What is the least common multiple of 15, 18, and 24? 1 12 Apr 2016, 01:24
9 Which of the following cannot be the least common multiple 6 06 Feb 2014, 02:24
21 If k is a common multiple of 75, 98, and 140, which of the 13 15 Oct 2013, 05:14
22 If the least common multiple of positive integer m and n is 13 12 Jun 2008, 23:09
Display posts from previous: Sort by

# For how many values of k is 12^12 the least common multiple

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

 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®.