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Difficulty:
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52% (01:27) correct
48%
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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
A. 4
B. 8
C. 10
D. 30
E. 60
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Updated on: 22 Sep 2023, 01:58
Originally posted by gmatophobia on 21 Sep 2023, 13:37.
Last edited by gmatophobia on 22 Sep 2023, 01:58, edited 1 time in total.
Last edited by gmatophobia on 22 Sep 2023, 01:58, edited 1 time in total.
Correcting silly errors :)
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Bunuel
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
General Discussion
