Least Common Multiple

The abbreviation LCM stands for "Least Common Multiple". The least common multiple of a number is the smallest number that is the product of two or more numbers. The least common multiple can be calculated for two or more integers as well as two or more fractions. The least common multiple of two numbers is the lowest possible number that can be divisible by both numbers. 

There is more than one method to find the LCM of two numbers. One of the quickest ways to find the LCM of two numbers is to use the prime factorization of each number and then the product of the least powers of the common prime factors will be the LCM of those numbers. Explore the world of LCM by going through its various aspects and properties. 

1.	What is Least Common Multiple (LCM)?
2.	How to find the Least Common Multiple?
3.	Formulas of LCM
4.	Relation Between LCM and HCF
5.	Difference Between LCM and HCF
6.	Properties of LCM
7.	FAQs on LCM

What is Least Common Multiple (LCM)?

The least common multiple is also known as "LCM" (or) the "Lowest Common Multiple" in Math. The least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers. Let’s take two numbers: say, 2 and 5. Each will have its own set of multiples. 

• Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
• Multiples of 5 are 5, 10, 15, 20, …

Let's represent these multiples on the number line and circle the common multiples,

Thus, the common multiples of 2 and 5 are 10, 20, ….. The smallest number among 10, 20, … is 10. So the least common multiple of 2 and 5 is 10. Therefore, LCM (2, 5) = 10

LCM of numbers can be calculated using various methods. There are 3 methods to find the least common multiple of two numbers. Each method is explained below with some examples of LCM.

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

LCM by Listing Method (Listing Out the Common Multiples)

By using the listing out the common multiples method we can find out the common multiples of two or more numbers. Out of these common multiples, the least common multiple is considered and the LCM of two given numbers can thus be calculated.  To calculate the LCM of the two numbers "A" and "B" by using listing out the common multiples follow the steps given below:

• Step 1 - List a few multiples of A and B
• Step 2 - Mark the common multiples from the multiples of both numbers.
• Step 3 - Select the smallest common multiple, that smallest common multiple is the LCM of the two numbers.

Example: Find the least common multiple (LCM) of 4 and 5. 

Solution:

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... and the multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ...

Hence, by the definition of least common multiple, the LCM of 4 and 5 is 20

LCM using Prime Factorization

By using the prime factorization method we can find out the prime factors of the numbers and these prime factors can be used to find the LCM of those numbers. To calculate the LCM of two numbers using the prime factorization method follow the steps given below:

• Step 1 - Represent the numbers in the prime factored form. 
• Step 2 - The LCM of the given two numbers is the product of all the prime factors. (However, common factors will be included only once) 

Let’s learn this method using the example given below.

Example: Find the least common multiple (LCM) of 60 and 90 using prime factorization. 

Solution:

• Step 1 - The prime factorization of 60 and 90 are: 60 = 2 × 2 × 3 × 5 and 90 = 2 × 3 × 3 × 5
• Step 2 - The product of all the prime factors = 2 × 2 × 3 × 5 × 3 = 180.

Therefore, LCM of 60 and 90 = 2 × 2 × 3 × 5 × 3 = 180

By Division Method

By division method, we will divide the numbers by a common prime number, and these dividends are used to calculate the LCM of those numbers. To calculate the LCM of two numbers using the division method follow the steps given below:

• Step 1 - Find a prime number which is a factor of at least one of the given numbers. Write this prime number on the left of the two numbers. For each number in the right column, do the following: 
• Step 2 - If the prime number in step 1 is a factor of the number, then divide the number by the prime and write the quotient below. If the prime number in step 1 is not a factor of the number, then write the number in the row below as it is. Continue the steps until all coprime numbers are left in the last row. (Two numbers are said to be coprime if they share no common factors other than 1.)

Let’s learn this method using the example given below.

Example: Find the least common multiple (LCM) of 6 and 15 using the common division method.

Solution:

• Step 1 - 2 is the smallest prime number and it is a factor of 6. Write this prime number on the left of the two numbers. For each number in the right column, continue finding out prime numbers which are their factors.
• Step 2 - 2 divides 6 but it's not a factor of 15, then write the number 15 in the row below as it is. Continue the steps until all coprime numbers are left in the last row. 
• Step 3 - The LCM is the product of all the prime numbers. LCM of 6 and 15 is, 2 × 3 × 5 = 30

Though we have three methods to find the LCM, the prime factorization method is the most common and easy method that we use.

LCM formulas are the collection of the numbers their LCM, HCF(Highest Common Factor), and their GCD(Greatest Common Divisor). These formulas are used to calculate the least common multiple of two integers as well as the LCM of two fractions. The LCM formulas for integers and fractions are shown below.

LCM Formula for Integers

If 'a' and 'b' are the two integers then the formula for their least common multiple is given as;

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

LCM Formula for Fractions

If 'a/b' and 'c/d' are the two fractions then the formula for their least common multiple is given as;

                LCM (a/b, c/d) = (LCM of Numerators)/(HCF of Denominators)

= LCM (a,c)/HCF (b,d)

HCF of two or more numbers is the highest common factor of the given numbers. It is calculated by multiplying the common prime factors of the given numbers. Whereas the least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers.
Let's assume 'a' and 'b' are the two numbers then the formula that gives the relationship between their LCM and HCF is given as:

LCM (a,b) × HCF (a,b) = a × b