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If the least common multiple of positive integers A and B is 160 and A

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If the least common multiple of positive integers A and B is 160 and A  [#permalink]

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New post 11 Feb 2019, 19:18
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00:00
A
B
C
D
E

Difficulty:

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Question Stats:

69% (01:58) correct 31% (01:53) wrong based on 77 sessions

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If the least common multiple of positive integers A and B is 160 and A:B=4:5, what is the greatest common divisor of A and B?

A) 8
B) 9
C) 10
D) 12
E) 15
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Re: If the least common multiple of positive integers A and B is 160 and A  [#permalink]

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New post 11 Feb 2019, 21:47
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From the ratio we can consider A=4x and B=5x.
GCD(A,B)=x

Property of LCM, GCD says, \(A*B=LCM(A,B)*GCD(A,B)\)
\(20x^2=160*x\)
\(20x=160\)
\(x=8\)
GCD(A,B)=8

Ans - A

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Re: If the least common multiple of positive integers A and B is 160 and A  [#permalink]

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New post 16 Feb 2019, 20:52
Official Answer

4 and 5 are relative prime numbers
A=4G
B=5G

L=4·5·G=160
20G=160, G=8
A
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Re: If the least common multiple of positive integers A and B is 160 and A  [#permalink]

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New post 17 Feb 2019, 23:24
philipssonicare wrote:
If the least common multiple of positive integers A and B is 160 and A:B=4:5, what is the greatest common divisor of A and B?

A) 8
B) 9
C) 10
D) 12
E) 15


Since LCM is 160, both numbers A & B together must have all the roots of 160.

Roots of 160 are = 5*2*2*2*2*2 = 5 * (2^5)

Now since A/B = 4/5 = (2^2)/5
=> A has 2^2 as additional factors over B
and B has 5 as additional factor A
(2^3) are the remaining roots between A and B which must b common.

=> A can be equal to 2^5
and B can be equal to (2^3)*5

Hence the GCD between the two is (2^3) = 8

Therefore option A is the correct answer.
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Re: If the least common multiple of positive integers A and B is 160 and A  [#permalink]

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New post 05 Mar 2019, 00:34
Hey,

Got it through the following equation:

LCM*GCF= AB
160*GCF= 5*A*4B
GCF=5*A*4B/160
GCF=AB/8

So GCF must be A since only 8/8 results in an integer. But what if the answer choice contains multiples of 8???
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Re: If the least common multiple of positive integers A and B is 160 and A  [#permalink]

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New post 05 Mar 2019, 01:30
iac00 wrote:
Hey,

Got it through the following equation:

LCM*GCF= AB
160*GCF= 5*A*4B
GCF=5*A*4B/160
GCF=AB/8

So GCF must be A since only 8/8 results in an integer. But what if the answer choice contains multiples of 8???


So from the question it's given that:

A:B = 4:5
=> A = 4x
B = 5x

Also since x will also be the greatest common divisor between the two; GCD = x

=> LCM * GCD = 4x * 5x = 20(x^2)
=> 160 * x = 20 (x^2)
=> x = 8

Hence we get a clear answer and we wont have a case of multiples.

Similar explanation has already been provided by GMATMBA5
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Re: If the least common multiple of positive integers A and B is 160 and A   [#permalink] 05 Mar 2019, 01:30
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If the least common multiple of positive integers A and B is 160 and A

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