How to Find LCM | Least Common Multiple, Definition, Meaning, Solved Problems

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? Book a Free Trial Lesson and match with top Math Tutors for concepts, homework help, and test prep.

What is the Least Common Multiple (LCM) in Maths?

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:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 26,…
• Multiples of 9: 9, 18, 27, 36,…

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.

Properties of LCM

There are certain properties of LCM that you should learn before knowing how to find LCM.

1. Associative Property of 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.

2. Commutative Property of LCM

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.

3. Distributive Property of LCM

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)

How to Find the LCM?

There are three major methods for finding the LCM of two or more numbers. The methods are:

1. Division Method

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:

• Step 1: Write all the given numbers for which you have to find the LCM, separated by commas.
• Step 2: Now, find the smallest prime number which is divisible for any of the given two numbers.
• Step 3: If any number is not divisible, write that number in the next row just below it and proceed further.
• Step 4: Continue dividing the numbers obtained after each step by the prime numbers, until you get the result as 1 in the entire row.
• Step 5: Now, multiply all the prime numbers and the final result will be the LCM of the given numbers.

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

2. Prime Factorization Method

To find the LCM of the given numbers using the prime factorization method, follow the steps given below:

• Step 1: Find the prime factors of the given numbers using the repeated division method explained above.
• Step 2: Write the prime factors in their exponent forms. Then multiply the prime factors having the highest power.
• Step 3: The final result after multiplication will be the LCM of the given numbers.

For example, you have to find the LCM of 18, 10, and 7 using the prime factorization method.

• Prime factorization of 18 can be expressed as 2 * 3 * 3 = 21 * 32
• Prime factorization of 10 can be expressed as 2 * 5 = 21 * 51
• Prime factorization of 7 can be expressed as 71

So, the LCM of 18, 10, and 7 = 2¹ * 3² * 5¹ * 7¹ = 2 * 3 * 3 * 5 * 7 = 630.

3. Listing Method

To find the LCM of the given numbers using the listing method, you can follow the following steps:

• Step 1: Write down the first few multiples of the given numbers separately.
• Step 2: Out of all the multiples of the numbers focus on the multiples which are common to all the given numbers.
• Step 3: Now, find out of all the common multiples, and take out the smallest common multiple. That will be the LCM of the given numbers

For example, you have to find the LCM of 8 and 5 using the listing method.

• Multiples of 8 are 8, 16, 24, 32, 40, 48, 64,…
• Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45,…

Here, it is clear that the least common multiple is 40.

So, the LCM of 8 and 5 is 40.

Important LCM Formulas

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).

1. Formula for finding the LCM of the given integers

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.

2. Formula for finding the LCM of the given fractions

LCM = LCM of the numerator / HCF of the denominator

LCM List

LCM Of 3 and 9

LCM Of 3 and 7

LCM Of 7 and 9

LCM Of 3 and 5

LCM Of 7 and 8

LCM Of 48 and 56

LCM of 120 and144

LCM of 4 and 10

LCM of 8 and 20

LCM of 6 and 10

LCM of 30 and 35

LCM of 3 and 8

LCM of 9 and 12

LCM of 4 and 6

LCM Of 80, 85, and 90

LCM Of 45, 60, and 75

LCM of 63, 70, and 77

Solved Example Problems Based on LCM

Looking to Learn Math? Book a Free Trial Lesson and match with top Math Tutors for concepts, homework help, and test prep.

FAQs on LCM

We hope this article on LCM is helpful to you.