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.

Answer: C.
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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.


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.

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

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:(

1. The answer is c
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

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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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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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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What is the value of integer x?

(1) The lowest common multiple of x and 16 is 48.

The prime factorization of 16 is 2^4. If the LCM common multiple of x and 16 is 48, this tells us x consists of a 3. However, we don't know what else the 3 contains. X could be 3, 6, 48, etc. Insufficient.

(2) The greatest common factor of x and 16 is 4

The GCF of x and 16 is 2^2. X could be 20, 28, etc. Insufficient.

(1&2) GCF * LCM = 192
192 / 16 = 12
X = 12

Answer is C.
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