Lowest Common Multiple – LCM

Least Common Multiple in maths is abbreviated as LCM and is used to find a number that is the smallest number that is divisible by two or more numbers perfectly. In other words, the LCM is the smallest multiple that each of the given numbers divides into evenly. LCM stands for Least Common Multiple i.e., LCM is the smallest multiple which is common for all the given numbers. We can easily find the LCM of two or more numbers by simply finding the prime factor of the given numbers and then taking the highest power of each factor of the numbers.

In this article, we will learn about Least Common Multiple (LCM) in maths, how to calculate LCM, its examples, and others in detail.

What is Least Common Multiple (LCM)?

Least Common Multiple or LCM in mathematics is defined as the smallest number which when divided by the given number gives the remainder zero. We can also say that the Least Common Multiple of any number is the number that contains all the multiples of the given number. We can explain the same using the example,

Example: Find the LCM of 4, 6.

Solution:

For finding LCM of 4, and 6. We write the multiple of 4 and 6 such that,

Multiple of 4 = 4, 8, 12, 14, 20, . . .

Multiple of 6 = 6, 12, 18, 24, 30, . . .

Here, comparing the multiples of 4, and 6 we find that the lowest number that is a multiple of both 4 and 6 is 12.

Thus, the LCM of 4 and 6 is 12

How to Find the LCM of two Numbers?

LCM or the Least Common Multiple is calculated by various methods and the most common three methods are,

• Finding LCM using Listing Method
• Finding LCM using Prime Factorization Method
• Finding LCM using Division Method

Finding LCM using Listing Method

Finding LCM using the listing method is achieved using the steps added below, suppose we have to find the LCM of two numbers P and Q then

Step 1: Find and list all the multiples of P and Q.

Step 2: Observe the multiples of P and Q and find the lowest common multiple among them.

Step 3: Now the smallest common multiple of these P and Q is the LCM of P and Q.

For example, find the LCM of 6 and 8.

Solution:

List the multiple of 6 and 8

• Multiples of 6 = 6, 12, 18, 24, 50, …
• Multiples of 8 = 8, 16, 24, 32, 40, …

The least common multiple of 6 and 8 is, 24

Thus, the LCM of 6 and 8 is 24

Finding LCM using Prime Factorization Method

In the prime factorization method, we write the prime factors of the numbers and then the factors with the highest exponent is the LCM of the number. Using the prime factorization method find the LCM of P and Q.

Step 1: Write the prime factors of P and Q.

Step 2: Write all the factors of P and Q in exponent form.

Step 3: Then we find the products of the factors with the highest exponents.

For example, find the LCM of 16 and 18.

Solution:

Write prime factors of 16, and 18

16 = 2×2×2×2 = 2⁴

18 = 2×3×3 = 2×3²

LCM of 16 and 18 = 2⁴×3²

= 16×9 = 144

Thus, the LCM of 16 and 18 is 144.

Finding LCM using Division Method

Division method is the method that is used to find the LCM of the number quickly. We find the LCM of two numbers 12 and 15 by following the steps added below,

Step 1: Write the number in the format as shown in the image added below,

Step 2: Starting from the first prime number write a prime number that is a factor of any of the number. (Here 2 is factor of 12). Divide the numbers with the prime to reduce the numbers.

Step 3: Continue the same pattern and then start increasing to next prime number as required. (Here, 2 is also a factor of 6 so we again divide by 2)

Step 4: Continue the Stpe 2 with next prime till we reduce all the numbers to 1

Step 5: Here, both the numbers are reduce to 1 so the LCM is the product of all the prime factors so obtained.

Thus, LCM of 12 and 15 = 2×2×3×5 = 60

Let’s consider another example for the same.

Example: Find the LCM of 20 and 30.

Solution:

LCM of 20 and 30 is calculated as,

LCM of 20 and 30 = 2×2×3×5 = 60

Here, 60 is the smallest multiple of 20 and 30 that is completely divided by both 20 and 30.

Least Common Multiple (LCM) Formula

We have the LCM formula that is used to find the LCM of two numbers if the HCF and the two numbers are given. The LCM formula states that the LCM of two numbers is calculated by diving the HCF of these two numbers by the product of two numbers, i.e. the LCM of two numbers P and Q is,

LCM (P, Q) = (P × Q) ÷ HCF(P, Q)

For example, find the LCM of 16 and 12 if its HCF is 4.

Solution:

12×16 = 192

HCF of 12, 16 = 4

LCM of 12, 16 = (12×16)/HCF of 12, 16

= 192/4

= 48

Thus, the LCM of 12 and 16 is 48.

Relationship Between LCM and HCF

The relation between the Least Common Multiple(LCM) and Highest Common Factor(HCF) is the relation that gives an expression defining the relation between LCM and the HCF, we know that the LCM of two numbers is the number that is the smallest multiple of the given two numbers and the HCF of two numbers is the highest factor of the two numbers. The relation between LCM and the HCF is,

Suppose we have two numbers (P and Q) then,

LCM (a, b) × HCF (a, b) = a × b

This is also stated as,

LCM of Numbers × HCF of Numbers = Product of two Numbers

Difference Between LCM and HCF

The difference between HCF and the LCM is shown in the table added below,

LCM (Least Common Multiple)	HCF (Highest Common Factor)
LCM of two numbers is defined as the smallest multiple of two numbers.	HCF of two numbers is the highest common factor of the two numbers.
For prime numbers, the LCM is the product of the number.	For prime numbers, the HCF is one(1).
For two numbers LCM is always greater than equal to the number itself.	For two numbers HCF is always smaller than equal to the number itself.
LCM of 15 and 20 is 60	HCF of 15 and 20 is 5

LCM of Three Numbers

We know that LCM or Least Common Multiple is calculated by taking the smallest that is the multiple of all the given numbers. Suppose we have to find the LCM of A, B, and C then we should follow the following steps,

Step 1: Find and list some of the multiples of the given three numbers A, B, and C.

Step 2: Observe the multiples of A, B, and C to find the lowest common multiple among them.

Step 3: Now the smallest common multiple of these A, B, and C is the LCM of A, B, and C.

Example: Find the LCM of 3, 4, and 6.

Solution:

List the multiple of 3, 4 and 6

• Multiples of 3 = 3, 6, 9, 12, 15, …
• Multiples of 6 = 6, 12, 18, 24, 30, …
• Multiples of 4 = 4, 8, 12, 16, 20, …

The least common multiple of 3, 4 and 6, is 12

Thus, the LCM of 3, 4 and 6 is 12.

LCM Formula

The Least Common Multiple (LCM) of two or more numbers is the smallest multiple that is exactly divisible by each of them. There is no specific formula for LCM, but it can be found using various methods like prime factorization or listing multiples. To find the LCM of numbers \(a\) and \(b\):

1. Prime Factorization Method: Write both numbers as products of their prime factors. LCM is the product of all unique prime factors, each raised to its highest power from the given numbers.

For example, LCM of 12 and 15:

2. Listing Multiples Method: List multiples of each number and identify the smallest common multiple.

For example, multiples of

Multiples of

Related Resources:

• HCF and LCM
• Percentage

Solved Examples on Least Common Multiple (LCM)

Example 1: Find the LCM of 8, 12, and 30 by the Prime Factorization Method.

Solution: 

• Prime Factors of 8 = 2 × 2 × 2 = 2³
• Prime Factors of 12 = 2 × 2 × 3 = 2² × 3
• Prime Factors of 30 = 2 × 3 × 3 × 5 = 2 × 3² × 5

LCM of 8, 12, and 30 = 2³× 3² × 5 = 360

Example 2: Find the LCM of 6, 8, and 16.

Solution:

List the multiple of 6, 8, and 16

• Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, …
• Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, …
• Multiple of 16 = 16, 32, 48, 64, ….

The least common multiple of 6, 8, and 16 is 48

Thus, the LCM of 6, 8, and 16 is 48.

Example 3: Find the LCM of 30 and 12 if its HCF is 6.

Solution:

As, 30×12 = 360, and

HCF of 30, 12 = 6

Thus, LCM of 12, 30 = (12×30)/HCF of 12, 30

⇒ LCM of 12, 30 = 360/6

⇒ LCM of 12, 30 = 60

Thus, the LCM of 12 and 30 is 30.

Practice Problems on Least Commmon Multiple (LCM)

1. Find the LCM of 12 and 18.

2. What is the LCM of 8, 10, and 15?

3. Calculate the LCM of 20 and 28.

4. Determine the LCM of 9, 12, and 15.

5. A school cafeteria offers pizza every 6 days and burgers every 9 days. If both pizza and burgers are offered today, when will they be offered on the same day again?

6. One city bus arrives at a stop every 15 minutes, while another arrives every 20 minutes. If both buses arrive at the stop simultaneously right now, how long will it be until they arrive at the same time again?

FAQs on Least Common Multiple (LCM)

1. Define Lowest Common Multiple with an example.

We define LCM or the Lowest Common Multiple of two numbers as the number that is lowest in value and is the multiple of both numbers. For example, the Lowest Common Multiple(LCM) of 5 and 15 is 15.

2. How to Find the Least Common Multiple (LCM)?

The Least Common Multiple (LCM) can easily be calculated using the given methods,

• Listing Method
• Prime Factorization Method
• Division Method

All these methods are discussed in the article.

3. What is the LCM of two Prime Numbers?

The LCM of two prime numbers is the product of the numbers.

4. What is Relation Between HCF and LCM?

The relation between HCF and LCM is, for two numbers ‘a’ and ‘b’.

“Product of a and b is equal to the product of HCF and LCM”, i.e.

LCM(a, b) × HCF(a, b) = a × b

5. What is the LCM of 25 and 75?

The LCM of 25 and 75 is 75.

6. What is the LCM of 12 and 8?

LCM of 8 and 12 is 24.

7. What is the LCM of 24 and 36?

LCM of 24 and 36 is 72.

8. Can the LCM of two numbers be smaller than both numbers?

No, the LCM of two numbers is always greater than or equal to the larger of the two numbers.

9. Can the LCM of a set of numbers be zero?

No, the LCM of a set of numbers is always a positive integer.

10. Is the LCM commutative?

Yes, the LCM is commutative, meaning that LCM(a, b) = LCM(b, a).

11. Can the LCM of two numbers be equal to one of the numbers?

Yes, if one of the numbers is a factor of the other, then the LCM can be equal to the larger number.

12. What is the Least Common Multiple Calculator?

Least Common Multiple (LCM) calculator is a tool or mathematical utility that calculates the least common multiple of two or more numbers. The LCM of two or more numbers is the smallest multiple that is evenly divisible by each of those numbers.