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# A set consists of three positive integers that are not coprime to each

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A set consists of three positive integers that are not coprime to each [#permalink]
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Bunuel wrote:
What about if the numbers are 2*3, 2*5, and 3*5?

arrgg ! Yes, this is a valid case. I was blind slighted by the fact that all the numbers should share the same common factor. That's not the case

In that case, the LCM will be 30

Great questions as always ..
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A set consists of three positive integers that are not coprime to each [#permalink]
gmatophobia could you please explain your reasoning a bit more ? what is the common factor bewteen 6 ( 3*2 ), 15 ( 3*5 ) and 10 ( 2*5 ). if they are not co-prime to each other, there should be one common factor present in all 3 numbers, am I wrong ? But a common factor is present only in 2 of the three numbers.
Please correct my reasoning. Thank you
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Re: A set consists of three positive integers that are not coprime to each [#permalink]
gmatophobia wrote:
Bunuel wrote:
A set consists of three positive integers that are not coprime to each other. If none of the numbers is a multiple of the other, what is the minimum value of the least common multiple of these three numbers?

A. 4
B. 8
C. 10
D. 30
E. 60

Few constraints to take note of -

1) None of the numbers are co-primes. ⇒ Inference: The numbers share at least common factor
2) None of the numbers is a multiple of the other ⇒ Inference: The numbers do not share more than one common factor

Combining the information: The three numbers share only one common factor. None of the three numbers is 1.

Question: Minimum value of the least common multiple of these three numbers

Two approaches to solve this -

1) Logical Reasoning

For the LCM to have the least value, the constituent prime number should be least. The LCM will comprise three prime numbers, out of which one will be common to each of the numbers, and the other two will be associated with the two numbers. As we need the minimum value of LCM, the prime numbers should be as small as possible.

The first three prime numbers are 2, 3, and 5. We can select two prime numbers from these three available prime numbers to form (2 * 3), (3 * 5), (2 * 5). The LCM in that case will be 2 * 5 * 3 = 30

2) Using Options

As the options are already ordered, let's start with the lowest one -

A. 4

4 = 2 * 2

We just have two prime numbers. We cannot form three numbers without violating the constraints.

B. 8

8 = 2 * 2 * 2

The three numbers should share only one common factor. As there is no other prime number available, we cannot cannot form three numbers without violating the constraints.

C. 10

10 = 2 * 5

We need at least three prime numbers. Hence, we can eliminate this option.

D. 30

30 = 2 * 3 * 5

We have three prime numbers in this option. From the three available prime numbers, we can select two prime numbers so as to form three pairs. Each of the three pairs will have one common factor with one another, hence none of the numbers are co-primes. Also, as each pair consists of one prime number that's different the numbers are not multiple of one another.

The numbers are (2*3), (3*5), and (5*2). Hence 30 is a valid value of LCM.

E. 60

Option D

­How can 2,3,5 be non co-prime numbers as inference drawn. All 3 are co-prime as all share only 1 common factor. Is the question statement wrong? Please help
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Re: A set consists of three positive integers that are not coprime to each [#permalink]
Bunuel
I proceeded with the numbers being 4, 6 and 10 so the LCM was 60. I understand choosing 6, 15 and 10 gives a lesser LCM but i don't get the logic behind it.

What is the approach to come up with 6, 15 and 10?
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Re: A set consists of three positive integers that are not coprime to each [#permalink]
KarishmaB Bunuel Can someone please explain how to approach this type of question?
How to come up with 6, 15 and 10?
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Re: A set consists of three positive integers that are not coprime to each [#permalink]
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Bunuel wrote:
A set consists of three positive integers that are not coprime to each other. If none of the numbers is a multiple of the other, what is the minimum value of the least common multiple of these three numbers?

A. 4
B. 8
C. 10
D. 30
E. 60

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­The LCM should be lowest so the numbers should have minimum and smallest unique factors. Take, 2,3 and 5, the three smallest factors. Their LCM is 30.
But the numbers should not be coprime to each other and should not be multiple of each other.
Hence each number must have a factor common with the other two and a factor different from the other two.
2*3
3*5
2*5

Each number has a common factor with the other two and a factor different from the other two. Hence the numbers are 6, 15 and 10 and their LCM is 30.