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

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For how many values of k is 12^12 the least common multiple [#permalink] New post 12 Nov 2009, 03:38
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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, 00:30, edited 1 time in total.
Edited the question and added the OA
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Re: least common multiple -12^12 [#permalink] New post 12 Nov 2009, 04:12
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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.
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Re: least common multiple -12^12 [#permalink] New post 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.
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1) Translating the English to Math : word-problems-made-easy-87346.html
2) 'Work' Problems Made Easy : work-word-problems-made-easy-87357.html
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Re: least common multiple -12^12 [#permalink] New post 12 Nov 2009, 04:42
Expert's post
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.
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Re: least common multiple -12^12 [#permalink] New post 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} !! :wink:
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compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

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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

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Re: least common multiple -12^12 [#permalink] New post 12 Nov 2009, 08:43
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
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Re: least common multiple -12^12 [#permalink] New post 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!
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Re: least common multiple -12^12 [#permalink] New post 30 Mar 2012, 00:29
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Expert's post
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.
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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.


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Re: least common multiple -12^12 [#permalink] New post 30 Mar 2012, 03:43
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Expert's post
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
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Re: For how many values of k is 12^12 the least common multiple [#permalink] New post 31 Mar 2012, 11:48
ahhhhh I see it now; thanks so much bunuel & karishma, that clarified it
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Re: For how many values of k is 12^12 the least common multiple [#permalink] New post 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?
Re: For how many values of k is 12^12 the least common multiple   [#permalink] 23 Aug 2014, 09:21
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