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What is the value of integer x?

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What is the value of integer x?  [#permalink]

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Updated on: 28 Jan 2017, 05:07
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What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48.
(2) The greatest common factor of x and 16 is 4

have no idea why the answer is not E,cause
from (1) i have ...6->12->24
from (2) i have ...4->12->24
thanks for explanation;)

this question from Veritas prep:)

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Originally posted by 3111987 on 21 Jan 2014, 14:48.
Last edited by stonecold on 28 Jan 2017, 05:07, edited 1 time in total.
Edited.
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Re: What is the value of integer x?  [#permalink]

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21 Jan 2014, 16:48
5
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What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48 --> x can be 3, 6, 12, 24, or 48. Not sufficient.

(2) The greatest common factor of x and 16 is 4 --> x can take more than one value, for example, 4, 12, 20, .. (basically any number of the form 4*odd). Not sufficient.

(1)+(2) The most important property of LCM and GCD is: for any positive integers $$x$$ and $$y$$, $$x*y=GCD(x,y)*LCM(x,y)$$. According to this 16x=48*4 --> x=12. Sufficient.

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Re: What is the value of integer x?  [#permalink]

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Updated on: 22 Jan 2014, 10:42
This question is straightforward as the factors are simple and easy to digest.
For more complex questions of the type, it helps to draw table with the prime factors.

4 = 2^2
16 = 2^4
48 = 2^4 * 3

With that in mind consider the following two tables showing the overlaps for GCF and LCM.
Attachment:

image.png [ 15.7 KiB | Viewed 8287 times ]

Statement (1)
The prime factors for 16 are clear, they only include 2. So if the LCM is 48, the 3^1 must come from x.
However, x could also include some "2"s. If it includes none, then it would be 2^0 for x. Or just one 2.
Then it would be 2^1. In both these cases the still LCM would be 48. Statement (1) does not allow to clearly
answer what the value of x is. We only know it has a 3 which is not in 16. We don't know what else it has
as prime factors.

Statement (2)
Looking at the table again, now for GCF, we see the overlap is 2^2. We don't know the value of x. But
we are told the GCF of both numbers (x and 16) is 4.
But we don't know if x has any other prime factors. X could have many other prime factors that are not part of
16, thus other prime factors than 2. So Statement (2) is not sufficient.

Putting both statements together, we know x includes 2^2 (we know this from the GCF) and we know it
has a 3^1 (which is not part of 16 but in the LCM). Thus, x must be = 2^2*3^1.

Again ...this approach is "overdoing it" as the numbers are straightforward. But the moment the numbers get
more complicated or there are more than two numbers, it helps to structure your thoughts with such a table.

Originally posted by BabySmurf on 22 Jan 2014, 10:34.
Last edited by BabySmurf on 22 Jan 2014, 10:42, edited 2 times in total.
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Re: What is the value of integer x?  [#permalink]

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22 Jan 2014, 10:35
Bunuel wrote:
What is the value of integer x?
The most important property of LCM and GCD is: for any positive integers $$x$$ and $$y$$, $$x*y=GCD(x,y)*LCM(x,y)$$.

One thing I have always wondered to know, does this rule apply for multiple numbers or only when you have two numbers? Thanks.
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Re: What is the value of integer x?  [#permalink]

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22 Jan 2014, 15:24
Bunuel wrote:
What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48 --> x can be 3, 6, 12, 24, or 48. Not sufficient.

(2) The greatest common factor of x and 16 is 4 --> x can take more than one value, for example, 4, 12, 20, .. (basically any number of the form 4*odd). Not sufficient.

(1)+(2) The most important property of LCM and GCD is: for any positive integers $$x$$ and $$y$$, $$x*y=GCD(x,y)*LCM(x,y)$$. According to this 16x=48*4 --> x=12. Sufficient.

this is the best explanation I have heard so far) even my tutor from Veritas prep can't solve it today:(
thinking maybe i need to change my tutor?:)
btw-it so sad that you don't provide tutoring:(
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Re: What is the value of integer x?  [#permalink]

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22 Jan 2014, 15:53
3111987 wrote:
What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48.
(2) The greatest common factor of x and 16 is 4

have no idea why the answer is not E,cause
from (1) i have ...6->12->24
from (2) i have ...4->12->24
thanks for explanation;)

this question from Veritas prep:)

LCM(x,y) *GCM(x*y) =xy
so 48*4 = 16 *x
on solving x=12.
now you can cross verify
LCM(12,16) =48
GCM(12,16)=4
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Re: What is the value of integer x?  [#permalink]

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23 Jan 2014, 03:12
BabySmurf wrote:
Bunuel wrote:
What is the value of integer x?
The most important property of LCM and GCD is: for any positive integers $$x$$ and $$y$$, $$x*y=GCD(x,y)*LCM(x,y)$$.

One thing I have always wondered to know, does this rule apply for multiple numbers or only when you have two numbers? Thanks.

Check a simple example:

GCD(6, 10, 12) = 2;
LCM(6, 10, 12) = 60;

6*10*12 = 720, while GCD(6, 10, 12)*LCM(6, 10, 12) = 120.

Hope it's clear.
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Re: What is the value of integer x?  [#permalink]

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18 Jul 2015, 06:00
Bunuel wrote:
What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48 --> x can be 3, 6, 12, 24, or 48. Not sufficient.

(2) The greatest common factor of x and 16 is 4 --> x can take more than one value, for example, 4, 12, 20, .. (basically any number of the form 4*odd). Not sufficient.

(1)+(2) The most important property of LCM and GCD is: for any positive integers $$x$$ and $$y$$, $$x*y=GCD(x,y)*LCM(x,y)$$. According to this 16x=48*4 --> x=12. Sufficient.

Hello Bunuel,

Could you explain this part in red please? Why 4*odd?

Thank you.
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Re: What is the value of integer x?  [#permalink]

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19 Jul 2015, 03:41
pacifist85 wrote:
Bunuel wrote:
What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48 --> x can be 3, 6, 12, 24, or 48. Not sufficient.

(2) The greatest common factor of x and 16 is 4 --> x can take more than one value, for example, 4, 12, 20, .. (basically any number of the form 4*odd). Not sufficient.

(1)+(2) The most important property of LCM and GCD is: for any positive integers $$x$$ and $$y$$, $$x*y=GCD(x,y)*LCM(x,y)$$. According to this 16x=48*4 --> x=12. Sufficient.

Hello Bunuel,

Could you explain this part in red please? Why 4*odd?

Thank you.

Since the greatest common factor of x and 16 = 2^4 is 4 = 2^2, then x cannot have 2 in higher power than 2, hence 4*odd.
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Re: What is the value of integer x?  [#permalink]

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28 Jan 2017, 05:05
Excellent Question.
Here is what i did in this one -->
We need the value of integer x.
Statement 1-->
LCM(x,16)=48
Hence the possible values of x -->
3
3*2
3*2^2
3*2^3
3*2^4
Hence not sufficient.
Statement 2->
GCD(x,16)=4
Hence the general Expression for x will be => x=2^2*k where x is any odd number.
Hence not sufficient.

Combing the two statements->
x must be 3*2^2
Hence sufficient.

Hence C.

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Re: What is the value of integer x?  [#permalink]

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