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The LCM of three numbers is four times their GCF. Which of

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The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 12 Mar 2013, 00:36
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The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be same.
III. At least one number is the same as GCF.

A. I only
B. III only
C. I and III only
D. I and II only
E. II and III only

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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 12 Mar 2013, 21:29
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avinashrao9 wrote:
Hi Bunuel,

Is there any general approach to solve this problem??
Finding a set of 3 numbers might be difficult somtimes :(



Consider a generic approach.
LCM - L
GCF - G
L = 4*G

The numbers must be of the form: Gn, Gm, Gk.
Since other than G, L contains only 4, n, m and k must be of one of the three numbers: 1, 2, 4

Mind you, there must be one number which is G because otherwise, the GCF will change. GCF of 2G, 2G, 4G will be 2G, not G.
There must be one number which is 4G because otherwise, the LCM will change. LCM of G, 2G, 2G is 2G, not 4G

So the three numbers could be of the form:
G, 2G, 4G or
G, G, 4G or
G, 4G, 4G

If G is even, no number will be odd.
Hence I and II needn't be true. III must be true.

Answer (B)
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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 12 Mar 2013, 06:42
emmak wrote:
The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be same.
III. At least one number is the same as GCF.

A. I only
B. III only
C. I and III only
D. I and II only
E. II and III only


Consider the following set: {2, 4, 8}.

GCF = 2.
LCM = 8 = 4*2.

The set discards options I and II. Only B fits.

Answer: B.
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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 12 Mar 2013, 11:28
Hi Bunuel,

Is there any general approach to solve this problem??
Finding a set of 3 numbers might be difficult somtimes :(

Bunuel wrote:
emmak wrote:
The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be same.
III. At least one number is the same as GCF.

A. I only
B. III only
C. I and III only
D. I and II only
E. II and III only


Consider the following set: {2, 4, 8}.

GCF = 2.
LCM = 8 = 4*2.

The set discards options I and II. Only B fits.

Answer: B.
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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 27 Mar 2013, 14:09
VeritasPrepKarishma wrote:
avinashrao9 wrote:
Hi Bunuel,

Is there any general approach to solve this problem??
Finding a set of 3 numbers might be difficult somtimes :(



Consider a generic approach.
LCM - L
GCF - G
L = 4*G

The numbers must be of the form: Gn, Gm, Gk.
Since other than G, L contains only 4, n, m and k must be of one of the three numbers: 1, 2, 4

Mind you, there must be one number which is G because otherwise, the GCF will change. GCF of 2G, 2G, 4G will be 2G, not G.
There must be one number which is 4G because otherwise, the LCM will change. LCM of G, 2G, 2G is 2G, not 4G

So the three numbers could be of the form:
G, 2G, 4G or
G, G, 4G or
G, 4G, 4G

If G is even, no number will be odd.
Hence I and II needn't be true. III must be true.

Answer (B)

Hi karishma,
I could not understand anything out of this explanation. Could you please explain it to me in a more detailed manner??
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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 25 Dec 2013, 16:34
Bunuel wrote:
emmak wrote:
The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?

I. At least one of the numbers is odd.
II. Two of the three numbers must be same.
III. At least one number is the same as GCF.

A. I only
B. III only
C. I and III only
D. I and II only
E. II and III only


Consider the following set: {2, 4, 8}.

GCF = 2.
LCM = 8 = 4*2.

The set discards options I and II. Only B fits.

Answer: B.


Is there any algebraic/conceptual approach to this question Bunuel?

Thanks
Cheers!

J :)
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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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New post 25 Dec 2013, 18:28
1
Jlgdr,

I guess what Karishma is saying is this :

G*N, G*M, G*K are the 3 numbers whose GCF is G. LCM of these 3 numbers can be written as
LCM(G*N,G*M,G*K) = G*LCM(N,M,K) = 4 *G (given). So LCM(N,M,K) = 4. This would mean {N,M,K} can be {1,1,4} or {1,2,4} or {1,4,4} since the LCM of the 3 numbers is 4.
So the only possible solution is number B.
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Re: The LCM of three numbers is four times their GCF. Which of  [#permalink]

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