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# Two positive integers A and B, have a LCM of 120. What is

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Two positive integers A and B, have a LCM of 120. What is [#permalink]  05 Nov 2004, 01:26
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Two positive integers A and B, have a LCM of 120. What is the Greatest common factor of A & B.
(a) 20
(b) 18
(c) 15
(d) 12
(e) 10
Director
Joined: 31 Aug 2004
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I think it is 20.

If available I would have answered 120...
Senior Manager
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Having troubles with this one..

Let's say the numbers are: A=30, B=40.
LCM = 120, GCF = 10.

Now let's sayt the numbers are: A=8, B=15.
LCM=120. GCF =1 ?

Do i miss something here?
Director
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twixt, how 20? Unable to get this.

LCM = 120 = 2^3*3*5

Ans choices:
20 = 2^2*5
18 = 2*3^2 (I can rule out this..because if GCD is 2*3^2, 3^2 would have been in LCM)

15=3*5
12=2^2*3
10=2*5. Why not these other 4 choices other than the 18 one. I am unable to understand. Hope I am not missing something fundamental here.

OR should the question read, what CNAT be a GCD? - In which case 18 is the answer. I am sorry, I am completely bold on this one.
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venksune wrote:
twixt, how 20? Unable to get this.

LCM = 120 = 2^3*3*5

Ans choices:
20 = 2^2*5
18 = 2*3^2 (I can rule out this..because if GCD is 2*3^2, 3^2 would have been in LCM)

15=3*5
12=2^2*3
10=2*5. Why not these other 4 choices other than the 18 one. I am unable to understand. Hope I am not missing something fundamental here.

OR should the question read, what CNAT be a GCD? - In which case 18 is the answer. I am sorry, I am completely bold on this one.

Its a typo frm myside.
The question is :

Two positive integers A and B, have a LCM of 120. Which is not possibly the Greatest common factor of A & B.
(a) 20
(b) 18
(c) 15
(d) 12
(e) 10
GMAT Club Legend
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By the way, answer to the original question should be 60
What you do is find the prime factors of 120 which are:
2^3 * 3 * 5
To find the GCD, you remove ONE TIME the smallest prime which is 2.
You then have 2 numbers, 120 and 60 and the GCD is 60 where their LCM is 120
Let's take another example. The LCM of two numbers is 315. Prime factors are 3^3 * 5 * 7
Remove ONE TIME the smallest prime which is 3.
You then have 2 numbers: 315 and 105--> The greatest possible GCD is 105
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Best Regards,

Paul

Manager
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Paul,

Maybe I am missing something, but let's take another example.

Say the 2 numbers are 16 and 24.

16: 2^4
24: 2^3 * 3

LCM is 48. So we want to find the GCF of the two numbers having this LCM.

Following the way you did it, 48 = 2^4 * 3, remove ONE TIME the smallest prime, and we have 2^3 * 3 = 24. Now, you've 2 numbers: 24 and 48, and the GCF = 24.

But the GCF of the two original numbers (16 and 24) are 8, and not 24!
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Hi Dan, you are misinterpreting what I meant
Now, what you did was taking the two numbers 16 and 24 as given. What I did was given a LCM, find 2 numbers, any numbers, so that you will get the largest possible GCF.
Let's take another example
Given a LCM of 4375 for two integers a and b, what is the greatest possible GCF?
4375 = 5^4 * 7
The greatest possible GCF of a and b is found by removing the 1 time the smallest prime which is 5.
So a=4375 and b=875 or vice versa. For all you know, the GCF will be 875.
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Best Regards,

Paul

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I have this formula :

GCF * LCM = Product of the numbers

In this case LCM = 120

so GCF * 120 = A*B

Lets put the values in -

A* B = 20 * 120 , so A can be 20, B can be 120
A* B = 18 * 120 , need to evaluate
A* B = 15 * 120 , A can be 15 and B can be 120
A* B = 12 * 120 , A can be 12 and B can be 120
A* B = 10 * 120 , A can be 10 and B can be 120

Since all the choices are easily eliminated we are left with 18, so this must be the answer. However we can further check

A* B = 18 * 120

IF GCF is 18, so both numbers must be multiple of 18 ie 18, 36, 54, 72, 90, 108, 126 and so on..... They will never make an LCM of 120.
Senior Manager
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Quote:
To find the GCD, you remove ONE TIME the smallest prime which is 2.
You then have 2 numbers, 120 and 60 and the GCD is 60 where their LCM is 120
Let's take another example. The LCM of two numbers is 315. Prime factors are 3^3 * 5 * 7
Remove ONE TIME the smallest prime which is 3.
You then have 2 numbers: 315 and 105--> The greatest possible GCD is 105

You guyz are genius. Thank You.
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