LCM Formula | List of LCM Formula You Should Know - BYJUS

# LCM Formula

LCM is a method of finding the least common multiple of two or more numbers. The value of the LCM is used to optimize the quantities of the given values. In this article, we will learn how to find the lowest common multiple of two or more numbers....Read MoreRead Less

### What is the LCM Formula?

The least common multiple (LCM) of two or more numbers is the product of their factors, using each common prime factor only once. When two or more numbers share the same multiples, then those multiples are called common multiples. The numbers can have many common multiples but the number that is the smallest of all multiples is called the ‘Least Common Multiple’ or ‘Lowest Common Multiple’. So, the formula to find the LCM of two or more numbers is as follows,

$$LCM~(a, b) = \frac{\left | ~a.b~ \right |}{GCD~(a,~b)}$$

Now, let’s see what each term indicates in this expression,

a = Number a

b = Number b

LCM(a, b) = Least common multiple of numbers a and b

GCD(a, b) or GCF(a, b) = Greatest common divisor/factor of numbers a and b

[Note: The Greatest Common Factor (GCF) or GCD of two or more numbers should be known in order to calculate the LCM. We can obtain the GCF of the given numbers by multiplying all the common factors of the numbers.]

### Rapid Recall ### Solved Examples

Example 1:

Find the LCM of 12 and 15.

Solution:

Given, two numbers are a = 12 and b = 15.

The prime factorization of 12 and 15 is,

12 = 2 x 2 x 3

15 = 3 x 5

Since 3 is the only common factor in both the numbers, then

GCF(12, 15) = 3

Now, the LCM of 12 and 15 can be determined by using the formula,

$$LCM(a,~b)=\frac{\left | ~a.b~ \right |}{GCF(a,~b)}$$

$$LCM(12,~15)=\frac{\left | ~12\times 15~ \right |}{GCF(12,15)}$$

$$=\frac{\left | ~12\times 15~ \right |}{3}$$        [Put the values]

$$=\left | ~12\times 5~ \right |$$     [Dividing 15 by 3]

= 60

Therefore, the least common multiple of 12 and 15 is 60.

Example 2:

Determine the LCM of two numbers 150 and 30.

Solution:

Let a = 150 and b = 30

The prime factors for a and b are

150 = 2 x 3 x 5 x 5

30 = 2 x 3 x 5

As we know, the product of all the common factors gives the GCF of 30 and 150.

That is, 2 x 3 x 5 = 30

Now, the formula to find the lowest common multiple is,

$$LCM(a,~b)=\frac{\left | ~a.b~ \right |}{GCF(a,b)}$$

$$LCM(30,~150)=\frac{\left | ~30~\times~ 150~ \right |}{GCF(30,\ 150)}$$     [Put the values]

$$=\frac{\left | ~30~\times ~150 ~\right |}{30}$$                                  [Dividing 30 by 30]

= 150

Hence, the LCM of 150 and 30 is 150.

Example 3:

The bell at St. Albert’s School is programmed to ring every 50 minutes for students in grades 1 to 5. While another bell will ring once every hour for students in grades 6 to 10. How many times in the next 10 hours will both the bells ring simultaneously? Solution:

According to the given information, let ‘a’ be the bell for the grades 1 to 5 that rings for every 50 mins.

And the bell ‘b’ is for grades 6 to 10 that rings for every 1 hr (60 mins).

In order to find how many times in the next 12 hours both the bells ring, we have to find the LCM of both the bells.

Then, let’s find the prime factors of 50 and 60.

50 = 2 x 5 x 5

60 = 2 x 2 x 3 x 5

The product of the common factors of 50 and 60 are 5 x 2 = 10.

Thus, 10 is the GCF of 50 and 60.

The LCM of 50 and 60 can be find with the formula,

$$LCM(a,~b)=\frac{\left | ~a.b~ \right |}{GCF(a,~b)}$$

$$LCM(50,~60)=\frac{\left | ~50\ \times\ 60~ \right |}{GCF(50,\ 60)}$$

$$=\frac{\left | ~50\ \times\ 60~ \right |}{10}$$     [Dividing 60 by 10]

$$=\left | ~50\ \times\ 6~ \right |$$

$$=300$$

Time taken for both the bells to ring together = 300 minutes

$$= \frac{300}{60}$$         [Convert minutes into hours]

$$= 5$$ hours

So, the number of times both the bells ring together = $$\frac{10\ \text{hours}}{5\ \text{hours}}=2$$