Tutoring
Tutors by Subject
Computer Science
Math
AP (Advanced Placement)
Courses
Coding Classes for Kids
Robotics Classes for Kids
Design Classes for Kids
Resources
AP (Advance Placement)
Calculators
Length Calculators
Weight Calculators
Tools
Tutorials
Scratch Tutorial
Learn
Math
AP (Advanced Placement)
Blog
Contents
The full form of LCM is the Least Common Multiple. In mathematics, the least common multiple (LCM) is a method to find the lowest possible common number of two numbers that are divisible by both numbers. LCM can be calculated for two or more numbers. LCM is also commonly referred to as the Least Common Divisor (LCD).
There are three methods for determining the LCM of a given number: the listing method, prime factorization method, and division method. In this article, let us discuss everything about how to find the LCM of a given number with solved examples. Scroll down to find out more.
Looking to Learn Math? Explore Wiingy’s Online Math Tutoring Services to learn from top mathematicians and experts.
The smallest positive number that is a multiple of two or more numbers.LCM Definition
For example, the LCM of 4 and 9 is 2 * 2 * 3 * 3 = 36.
Here, 4 is expressed as 2 * 2, and 9 is expressed as 3 * 3.
If we consider the multiples of 4 and 9, we get:
The first common multiple for 4 and 9 is 36. Hence, the LCM of 4 and 9 is 36.
LCM is used for the addition and subtraction of two fractions. When the denominator value of the fractions is not the same, LCM is used to make the denominators equal. This makes the entire calculating process easy.
There are certain properties of LCM that you should learn before knowing how to find LCM.
The associative property of LCM states that the LCM of A and B will be the same as the LCM of B and A.
LCM (A, B) = LCM (B, A)
For example, let’s consider A as 6 and B as 2. Here, 6 can be expressed as 2 * 3, and 2 can be expressed as 2.
So, 6 * 2 = 2 * 3 * 2.
Now, we will write the numbers in the exponent form and multiply the factors that have the highest power.
On writing the number in exponent form, we get:
6 = 2 * 3 = 21 * 31
2 = 21
So, the LCM of 6 and 2 is 21 * 31 = 6
And, LCM of 2 and 6 is also 6.
Hence, it is proved that LCM (6, 2) = LCM (2, 6) = 6.
The commutative property is used while dealing with finding the LCM of 3 numbers. The commutative property of LCM states that,
LCM (A, B, C) = LCM (LCM (A, B), C) = LCM (A, LCM (B, C))
For example, let’s consider A as 3, B as 6, and C as 12. Here, 3 can be expressed as 3, 6 can be expressed as 2 * 3, and 12 can be expressed as 2 * 2 * 3.
Now, we will write the numbers in the exponent form and multiply the factors that have the highest power.
On writing the number in exponent form, we get:
3 = 31
6 = 2 * 3 = 21 * 31
12 = 2 * 2 * 3 = 22 * 31
So, the LCM of 3, 6, and 12 is 22 * 31 = 2 * 2 * 3 = 12
Now, the LCM of A and B, that is, LCM 3 and 6 is 31 * 21 = 6 and
LCM of (A, B) and C, that is, LCM of 6 and 12 is 22 * 31 = 2 * 2 * 3 = 12.
LCM (LCM (A, B), C) = LCM (LCM (3, 6), 12) = LCM (6, 12) = 12
LCM of B and C, that is, LCM of 6 and 12 is 22 * 31 = 2 * 2 * 3 = 12 and
LCM of A and LCM of (B, C), that is, LCM of 3 and 12 is 22 * 31 = 2 * 2 * 3 = 12.
LCM (A, LCM (B, C)) = LCM (3, LCM (6, 12)) = LCM (3, 12) = 12.
Hence, it is proved that LCM (3, 6, 12) = LCM (LCM (3, 6), 12) = LCM (3, LCM (6, 12)) = 12.
The distributive property is also used while dealing with finding the LCM of 3 numbers. Distributive property of LCM states that,
LCM (dA, dB, dC) = d * LCM (A, B, C)
For example, let’s consider A as 5, B as 8, C as 13, and d to be any random variable.
Now, we will write the numbers in the exponent form and multiply the factors that have the highest power.
On writing the number in exponent form, we get:
5 = 51
8 = 2 * 2 * 2 = 23
13 = 131
LCM of 5, 8, and 13 is 51 * 23 * 131 = 5 * 2 * 2 * 2 * 13 = 520.
So, LCM (5d, 8d, 13d) = d * LCM (5, 8, 13) = 520
Hence, it is proved that LCM (5d, 8d, 13d) = d * LCM (5, 8, 13)
There are three major methods for finding the LCM of two or more numbers. The methods are:
To find the LCM using the division method, divide the given numbers by the smallest prime number, which is divisible by any of the given numbers. Then, the prime factors further obtained will be used to calculate the final LCM.
You can follow the following steps to find the LCM using the division method:
For example, you have to find the LCM of 12 and 5 using the division method.
Prime Factors | First Number | Second Number |
2 | 12 | 5 |
2 | 6 | 5 |
3 | 3 | 5 |
5 | 1 | 5 |
1 | 1 |
So, LCM of 12 and 5 = 2 * 2* 3 * 5 = 50
To find the LCM of the given numbers using the prime factorization method, follow the steps given below:
For example, you have to find the LCM of 18, 10, and 7 using the prime factorization method.
So, the LCM of 18, 10, and 7 = 21 * 32 * 51 * 71 = 2 * 3 * 3 * 5 * 7 = 630.
To find the LCM of the given numbers using the listing method, you can follow the following steps:
For example, you have to find the LCM of 8 and 5 using the listing method.
Here, it is clear that the least common multiple is 40.
So, the LCM of 8 and 5 is 40.
There are two major LCM formulas, one for finding the LCM of integers and the other for finding the LCM of fractions.
Before moving forward and knowing the formulas, you should know about the HCF (Highest Common Factor).
HCF is the highest factor which is common among the factors of all the given numbers. It is also known as the greatest common divisor (GCD).
Let A and B be two given integers. So, the LCM of A and B can be calculated using the formula:
LCM (A, B) = (A * B) / HCF (A, B),
where HCF is the highest common factor or the greatest common divisor of A and B.
Another formula for finding the LCM of the given integers is:
A * B = LCM (A, B) * HCF (A, B), that is,
The product of the two given integers is equal to the product of their LCM and HCF.
LCM = LCM of the numerator / HCF of the denominator
LCM List
Question 1: Find the LCM of 9 and 4 using the listing method.
Solution:
Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90,…
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40,…
Here, it is clear that the least common multiple is 36.
So, the LCM of 9 and 4 is 36.
Question 2: What is the LCM of 16 and 21 using the prime factorization method?
Solution:
Prime factorization of 16 can be expressed as 2 * 2 * 2 * 2 = 24
Prime factorization of 21 can be expressed as 3 * 7 = 31 * 71
So, the LCM of 16 and 21 = 24 * 31 * 71 = 2 * 2 * 2 * 2 * 3 * 7 = 336.
Question 3: If the LCM and HCF of two numbers, 5 and B, are 45 and 1, respectively. Find B.
Solution:
As we know,
Product of two numbers = LCM * HCF
We are given that,
One of the numbers = 5, LCM = 45, and HCF = 1
So, 5 * B = 45 * 1
B = (45 * 1) / 5
B = 9
Hence, the other number is 9.
Question 4: Find the LCM of 24 and 45 using the division method.
Solution:
Prime Factors | First Number | Second Number |
2 | 24 | 45 |
2 | 12 | 45 |
2 | 6 | 45 |
3 | 3 | 45 |
3 | 1 | 15 |
5 | 1 | 5 |
1 | 1 |
So, the LCM of 24 and 45 = 2 * 2 * 2 * 3 * 3 * 5 = 360.
Question 5: Find the LCM of 14, 22, and 18 using the prime factorization method.
Solution:
Prime factorization of 14 can be expressed as 2 * 7 = 21 * 71
I can express the prime factorization of 22 as 2 * 11 = 21 * 111
Prime factorization of 18 can be expressed as 2 * 3 * 3 = 21 * 32
So, the LCM of 14, 22, and 18 = 21 * 71 * 111 * 32 = 2 * 7 * 11 * 3 * 3 = 1386.
Question 6: If the HCF of two numbers, 42 and 9, is 3. Find the LCM.
Solution:
As we know,
LCM (A, B) = (A * B) / HCF (A, B)
We are given that,
Numbers = 42 and 9 and HCF = 3
So, LCM (42, 9) = (42 * 9) / HCF (42, 9)
= 378 / 3
= 126
Hence, the LCM of 42 and 9 is 126.
Looking to Learn Math? Explore Wiingy’s Online Math Tutoring Services to learn from top mathematicians and experts.
What do you mean by LCM?
LCM is the least common multiple. It is used to find the lowest possible common number that is divisible by all the numbers, for which you have to find the LCM.
How do LCM and HCF are connected?
LCM (A, B) = (A * B) / HCF (A, B),
where A and B are two integers.
This formula is used to find the LCM of the given integers.
Can LCM be calculated for only 2 numbers?
No, LCM can be calculated for more than two numbers as well. There should be at least 2 numbers for finding the LCM.
What are the methods to find the LCM?
There are three major methods for finding the LCM of two or more numbers. The methods are:
1. Division Method
2. Prime Factorization Method
3. Listing Method
What is the LCM of 2 and 8?
LCM of 2 and 8 is 8.
What are the properties of LCM?
There are three major properties of LCM. They are:
1. Associative property
2. Commutative property
3. Distributive property
What is the formula for finding the LCM of the fraction?
The formula for finding the LCM of a given fraction is:
LCM = LCM of the numerator / HCF of the denominator
We hope this article on LCM is helpful to you.
Practice Quiz
Questions: 1/2
Wrong Answer
Each charity would receive 30.2 dollars (151 divided by 5 = 30.2).