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

Question

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

KarishmaB · Accepted Answer

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)

Bunuel · Answer

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.

avinashrao9 · Answer

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

sauravleo123 · Answer

Hi karishma,
I could not understand anything out of this explanation. Could you please explain it to me in a more detailed manner??

jlgdr · Answer

Is there any algebraic/conceptual approach to this question Bunuel? Thanks Cheers! J

anjancgc · Answer

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.

ScottTargetTestPrep · Answer

If the three numbers are 4, 8 and 16, the LCM is 16 and the GCF is 4. We see that 16 is four times 4, but neither any of the numbers is odd nor two of them are the same. Thus, we see that I and II are not true. We see that III can be true since 4 is one of the three numbers. Without analyzing III further, by looking at the given answer choices, we can reject A, C, D and E; therefore, we are left with B as the correct answer.

Answer: B

puneetb · Answer

Can you elaborate this particular question through Venn diagram??

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

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