LCM Formula

LCM Formula: LCM stands for Least Common Multiple. LCM of two numbers say a and b is defined as the smallest positive integer divisible by both the numbers a and b. Hence, the LCM is the smallest common multiple of two or more numbers. It is also called lowest common multiple, or smallest common multiple.

In this article, we will discuss LCM, Formulas to calculate LCM, and different methods used to find the LCM of two or more numbers.

What is LCM?

LCM stands for Least Common Multiples which can be defined as the smallest positive integer that is evenly divisible by all the given numbers. It is also known as the Least Common Divisor (LCD). For example: LCM (5, 15) = 15, Here 15 is the smallest common multiple divisible by both 5 and 15.

LCM Formula

The table below represents the LCM formulas used in different cases:

Formulas of LCM
LCM of any two numbers	LCM (a,b) = [Tex]\frac {ab} {GCD(a,b)}[/Tex]
LCM of Fractions	LCM = [Tex]\frac {LCM of Numerator} {HCF of Denominator}[/Tex]

Finding LCM using HCF Formula

LCM can also be derived using GCD or HCF. If a, and b are any numbers then, we have,

LCM(a,b) × HCF(a,b) = a ×  b

or

LCM(a,b) =  a × b / GCD(a,b)

Note: This formula is only applicable while finding the LCM of two numbers only.

Example: Find LCM of 4, 56 using GCD of 4, 56.

Solution:

Prime factors of 4 = 2 × 2

Prime factors of 56 = 2 × 2 × 2 × 7

Common factor = 2 × 2 = 4

Hence,

GCD of 4, 56 = 4

LCM of 4, 56 = (4 × 56)/ gcd of (4, 56)

= 224/ 4

= 56

LCM Formula for 3 Numbers

To find the Least Common Multiple (LCM) of three numbers, you can use a formula that involves their greatest common divisor (GCD).

LCM(a,b)=GCD(a,b) ∣a×b∣​

LCM of Fractions

To find the LCM of two fractions we first compute the LCM of Numerators and GCD of the Denominators. Then, both these results will be expressed as a fraction. The formula to calculate LCM of two fractions is given below:

LCM =  LCM of Numerators / GCD of Denominators  

Example: Find the LCM of 6/7 and 5/4.

Solution: 

Numerators are 6, 5 and Denominators are 7, 4

We know that LCM of Fractions = LCM of Numerators / GCD of Denominators  

Then, LCM(6, 5) = 30 

and GCD(7, 4) = 1

Hence, LCM of 6/7 and 5/4 = 30/1 = 30.

LCM Calculator

Try out the following calculator to find LCM of two or more numbers.

*calculator

How to Find LCM (Lowest Common Multiple)?

Least Common Multiple or LCM of two or more numbers can be find by the following three methods:

• Listing out Multiples of Each Number
• Prime Factorization Method
• Division Method

The detailed explanation of each is given below:

LCM by Listing Multiples

In this method, we need to list the multiples of each number until at least one of the multiples appears on all the lists. Then, the LCM is the smallest number that is on all of the lists.

Example: Find the LCM of 6, 7, and 21 by listing multiples.

Solution:

LCM of 6, 7, 21

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70.

Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210.

Now, the smallest number that is common in all the lists is 42.

Hence, 

LCM(6, 7, 21) = 42

LCM using Prime Factorization Method

In this method, we need to write all numbers as a product of their prime factors. Then, LCM will be the product of the highest powers of all prime numbers.

Example: Find the LCM of 15 and 8 using Prime Factorization Method.

Solution:

To Find LCM of 15, 8

Prime factorization of 15 = 3 × 5

Prime factorization of 8 = 2 ×  2 × 2

Hence,

LCM = 2³ × 3 × 5 = 120

LCM using Division Method

First, we need to write all the numbers in a horizontal line separated by commas. Then, we divide all the given numbers by the smallest prime number. Then, we write the quotient and undivided numbers in a new line below the previous line.

We repeat this procedure until we come to the stage where no prime factor is common. The, we find the product of all divisors and the resultant number we get is the LCM. 

Example: Find LCM of 6, 8, 5, 4, 3 by Division method.

Solution:

LCM of 6, 8, 5, 4, 3 can be calculated as follow:

Hence the LCM(6, 8, 5, 4, 3) = 120

People Also Read:

• HCF and LCM
• Factors of a number

LCM Formula Examples

Example 1: Find out the LCM of 4 and 10.

Solution: 

we know that LCM(a, b) = a × b/ GCD(a, b)        

Here, a = 4 and b = 10

a × b = 4 × 10 = 40

GCD(a, b) = 2

Hence, LCM(16, 10) = 40 /2 = 20

Example 2: Calculate the  LCM of 14, 12, 7, and 8.

Solution: 

LCM of 14, 12, 7, 8 = 2 × 2 × 2 × 3 × 7

                               = 168

Hence, LCM(14, 12, 7, 8) = 168

Example 3: Find out the LCM for 8 and 24.

Solution:

Prime Factorization of 8 = 2 × 2 × 2

Prime Factorization of 24 = 2 × 2 × 2 × 3

LCM = 2 × 2 × 2 × 3= 24

Example 4: Find out the LCM of 36, 24.

Solution:

Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360 etc.

Multiples of 24 = 24, 48, 72, 96, 120, etc.

Common multiple = 72

Hence, LCM of 36 and 24 = 72

Example 5: Find the least number divided by 48 and 72, which leaves the remainder 9 in each.

Solution:

First we find the LCM of the two numbers we get,

Prime Factorisation of 48 = 2 × 2 × 2 × 2 × 3

Prime Factorization of 72 = 2 × 2 × 2 × 3 × 3

Therefore, LCM of the two numbers is 2 × 2 × 2 × 2 × 3 × 3 = 144.

The least number divided by 48 and 76 leaving remainder 9 is (144 + 9) = 153.

Practice Question on LCM Formula

Question 1: Find the LCM of 22, 25, 11 and 8 by division method.

Question 2: If HCF(12, 24) = 12. Find LCM(12, 24).

Question 3: Find the LCM of 2/3 and 3/4.

Question 4: Prove that HCF(24, 48) × LCM(24, 48) = 24 × 48.

Question 5: Find the least number divided by 36 and 24, that leaves the remainder 3 in each case.

FAQs on LCM Formula

What is Formula of LCM and HCF?

The Formula of LCM and HCF is :

Product of two numbers= LCM of two numbers × HCF of two numbers

What are Methods to Find LCM?

LCM can be calculated by the following three methods:

• Prime Factorization
• Listing out Common Multiples
• Division

How to Find Lowest Common Multiple?

We can easily find the Lowest Common Multiple of two or more numbers by listing the multiples of all the numbers and finding the smallest common multiple of each.

What is Formula for GCD to LCM?

The Formula for GCD to LCM is given by:

GCD(a,b)/ab = LCM(a,b)

Why is 24 LCM of 8 and 12?

24 is divisible by both 8 and 12. Also 24 is the first and smallest common multiple of 8 and 12, hence it is considered as LCM of 8 and 12.

What is the full form of LCM?

The full form of LCM is Least Common Multiple.