The term LCM stands for “Least Common Multiple,” which refers to the lowest possible number that is divisible by two or more given numbers. It is also known as the Lowest Common Multiple. Calculating the LCM can be done for two numbers or a set of numbers. There are various methods to determine the LCM of a given set of numbers, with one efficient approach being the use of prime factorization. By finding the prime factors of each number and taking the product of the highest powers of the common prime factors, we can obtain the LCM. Explore this article to learn more about how to find lowest common multiples of numbers and the different techniques involved.
What is Least Common Multiple (LCM)?
The Least Common Multiple (LCM) refers to the smallest positive integer that is divisible by two or more given numbers without leaving a remainder. In simpler terms, it is the smallest common multiple of the numbers. The LCM is often used in various mathematical operations, such as adding or subtracting fractions with different denominators or solving equations involving multiple numbers.
For Example: LCM of 8 and 12
To find the LCM (Least Common Multiple) of 8 and 12, we can use listing multiples method.
Multiples of 8: 8, 16, 24, 32, 40, …
Multiples of 12: 12, 24, 36, 48, 60, …
From the lists, we can see that the smallest common multiple of 8 and 12 is 24.
Therefore, the LCM of 8 and 12 is 24.
How to Find LCM?
To find the LCM (Least Common Multiple) of two or more numbers, you can use various methods. Here are two commonly used methods:
1. Prime Factorization Method:
Example: Find the LCM of 4 and 6.
Prime factorization of 4: 22
Prime factorization of 6: 21 * 31
To find the LCM, we take the highest powers of the prime factors:
Highest power of 2: 22
Highest power of 3: 31
LCM = 22 * 31 = 4 * 3 = 12
Therefore, the LCM of 4 and 6 is 12.
2. Listing Multiples Method:
-
- Write down the multiples of each number until you find a common multiple.
- Look for the smallest common multiple among the lists of multiples.
Example: Find the LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, …
Multiples of 6: 6, 12, 18, 24, 30, …
From the lists, we can see that the smallest common multiple of 4 and 6 is 12.
Therefore, the LCM of 4 and 6 is 12.
LCM Examples
Example 1: LCM of 12 and 18
Method 1: Prime Factorization
Prime factorization of 12: 22 * 31
Prime factorization of 18: 21 * 32
To find the LCM, we take the highest powers of the common prime factors:
Highest power of 2: 22
Highest power of 3: 32
LCM = 22 * 32 = 4 * 9 = 36
Therefore, the LCM of 12 and 18 is 36.
Method 2: Listing Multiples
Multiples of 12: 12, 24, 36, 48, 60, …
Multiples of 18: 18, 36, 54, 72, 90, …
From the lists, we can see that the smallest common multiple of 12 and 18 is 36.
Therefore, the LCM of 12 and 18 is 36.
Example 2: LCM of 12 and 15
Method 1: Prime Factorization
Prime factorization of 12: 22 * 31
Prime factorization of 15: 31 * 51
To find the LCM, we take the highest powers of the common prime factors:
Highest power of 2: 22
Highest power of 3: 31
Highest power of 5: 51
LCM = 22 * 31 * 51 = 4 * 3 * 5 = 60
Therefore, the LCM of 12 and 15 is 60.
Method 2: Listing Multiples
Multiples of 12: 12, 24, 36, 48, 60, …
Multiples of 15: 15, 30, 45, 60, 75, …
From the lists, we can see that the smallest common multiple of 12 and 15 is 60.
Therefore, the LCM of 12 and 15 is 60.
Example 3: LCM of 3 and 4
Method 1: Prime Factorization
Prime factorization of 3: 31
Prime factorization of 4: 22
To find the LCM, we take the highest powers of the prime factors:
Highest power of 2: 22
Highest power of 3: 31
LCM = 22 * 31 = 4 * 3 = 12
Therefore, the LCM of 3 and 4 is 12.
Method 2: Listing Multiples
Multiples of 3: 3, 6, 9, 12, 15, …
Multiples of 4: 4, 8, 12, 16, 20, …
From the lists, we can see that the smallest common multiple of 3 and 4 is 12.
Therefore, the LCM of 3 and 4 is 12.
Example 4: LCM of 12 and 16
Method 1: Prime Factorization
Prime factorization of 12: 22 * 31
Prime factorization of 16: 24
To find the LCM, we take the highest powers of the prime factors:
Highest power of 2: 24
Highest power of 3: 31
LCM = 24 * 31 = 16 * 3 = 48
Therefore, the LCM of 12 and 16 is 48.
Method 2: Listing Multiples
Multiples of 12: 12, 24, 36, 48, 60, …
Multiples of 16: 16, 32, 48, 64, 80, …
From the lists, we can see that the smallest common multiple of 12 and 16 is 48.
Therefore, the LCM of 12 and 16 is 48.
Example 5: LCM of 8 and 9
Method 1: Prime Factorization
Prime factorization of 8: 23
Prime factorization of 9: 32
To find the LCM, we take the highest powers of the prime factors:
Highest power of 2: 23
Highest power of 3: 32
LCM = 23 * 32 = 8 * 9 = 72
Therefore, the LCM of 8 and 9 is 72.
Method 2: Listing Multiples
Multiples of 8: 8, 16, 24, 32, 40, …
Multiples of 9: 9, 18, 27, 36, 45, …
From the lists, we can see that the smallest common multiple of 8 and 9 is 72.
Therefore, the LCM of 8 and 9 is 72.
FAQs on Least Common Multiple (LCM)
What does LCM stand for?
LCM stands for “Least Common Multiple.”
What is the definition of LCM?
The LCM is the smallest positive integer that is divisible by two or more given numbers without leaving a remainder.
How is LCM different from GCF/HCF?
While LCM focuses on finding the smallest common multiple of given numbers, GCF (Greatest Common Factor) or HCF (Highest Common Factor) focuses on finding the largest number that divides all the given numbers.
Can the LCM be smaller than the given numbers?
No, the LCM is always equal to or greater than the given numbers.
Can the LCM be zero?
No, the LCM is always a positive integer. Zero cannot be the LCM of any set of numbers.
Can the LCM be negative?
No, the LCM is always a positive integer. Negative numbers do not have a concept of being divisible or forming multiples.
Can the LCM be calculated for fractions or decimal numbers?
Yes, the LCM can be calculated for fractions or decimal numbers. The numbers can be converted to their equivalent forms and then the LCM can be found using the same principles as with whole numbers.
Can the LCM be used for more than two numbers?
Yes, the LCM can be calculated for any number of given numbers. The same methods apply.
Can the LCM be used in real-life situations?
Yes, the LCM has practical applications such as finding the least common multiple of time intervals, scheduling events, calculating resource distribution, and more.