What is the least common Multiple or LCM?
The least common multiple or LCM of a set of numbers is the smallest or the least positive integer that is divisible by all the numbers in the set.
We can determine the LCM of two or more numbers using:
- Listing multiples method
- Prime factorization method
What is the LCM of 8 and 12?
Let us use the methods to find the LCM of 8 and 12:
In the 1st method, we will list the multiples of both numbers.
Multiples of 8: 8, 16, 24, 32, 40…
Multiples of 12: 12, 24, 36, 48…
As you can see, the smallest common multiple of 8 and 12 is 24 and hence, it is the least common multiple.
In the 2nd method, we will use the prime factorization method to find the least common multiple of 8 and 12.
We will draw a factor tree for each number.
Now, we will write the prime factorization of each number and take the product of all the highest exponents of all factors for each number.
8 = 222
12 = 223
Among the prime factors the highest exponent of 2 appears in factorization of 8 and that for 3 appears in factorization of 12.
So, the LCM will be,
Product of prime factors = 2.2.2.3 = 24
So, the least common multiple of 8 and 12 is 24.
Solved Examples
Example 1: Margaret and John complete swimming laps in every 8th and 12th minute respectively. At what time will both of them complete their laps together?
Solution:
We can find out the common multiples of 8 and 12 to find the result of the problem.
As we know,
Multiples of 8 : 8, 16, 24, 32, 40…
Multiples of 12 : 12, 24, 36, 48…
Since 24 is the common multiple of 8 and 12, hence, at the 24th minute both Margaret and John will finish the lap together.
Example 2: Which is the smallest number that is exactly divisible by 8 and 12?
Solution:
The smallest number that is divisible by 8 and 12 exactly is their LCM.
Now, we can find the multiples of 8 and 12 by listing them.
Multiples of 8: 8, 16, 24, 32, 40…
Multiples of 12: 12, 24, 36, 48…
Since the LCM of 8 and 12 is 24, hence the smallest number that is divisible by 8 and 12 is 24.
Example 3: What is the GCF of 8 and 12?
Solution:
We know that the LCM of the numbers 8 and 12 is 24.
\(GCF \times LMC = 8 \times 12\) [Relation between GCF, LCM & the numbers in a set]
\(GCF \times 24 = 8 \times 12\) [Substitute the LCM value]
\(GCF = \text{ }4\) [Division Property of Equality]
Hence, the GCF of 8 and 12 is 4.
No, it is not. LCM of 8 and 12 is 24, and LCM of 4 and 8 is 8. Hence, they are not the same.
The relationship is that the product of GCF and LCM is equal to the product of the numbers in a given set of numbers.
We can find the LCM of a set of numbers by listing multiples and by following the prime factorization method.