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 factor2) None of the numbers is a multiple of the other ⇒
Inference: The numbers do not share more than one common factorCombining 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 ReasoningFor 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. The numbers are (2*2), (2*3), and (2*5). The LCM in that case will be (\(2^2\) * 3 * 5) = 60
2)
Using OptionsAs the options are already ordered, let's start with the lowest one -
A. 44 = 2 * 2
We just have two prime numbers. We cannot form three numbers without violating the constraints.
B. 88 = 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. 1010 = 2 * 5
We need at least three prime numbers. Hence, we can eliminate this option.
D. 3030 = 2 * 3 * 5
We have three prime numbers in this option. Let's try to create three numbers. First, the lowest prime number can be common across all three numbers. So each number has a 2 in its prime factorized form.
Second, one of the numbers can have 3 and the third number can have 5. So we have 2, (2*3) and (2*5). However, the question stems also mentions that "none of the numbers is a multiple of the other". In this case, 2 is a multiple of 6 and 10. Hence, we need to add another 2 to the first number.
E. 602 * 3 * 2 * 5
The numbers are (2 * 2), (2 * 3) and (2 * 5)
Option E