Least Common Multiple

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

There are multiple methods 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 highest powers of the common prime factors will be the LCM of those numbers.

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. It can be written as 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.

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

LCM by Listing Method

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 the listing method, follow the steps given below:

• Step 1: List the first 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 lowest 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 × 3 × 5 = 180.

LCM by Division Method

By the division method, we will divide the numbers by a common prime number, and these prime factors 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 given numbers.
• 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 1 is left in the last row.

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

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

Solution:

• Step 1: 2 is the smallest prime number and it is a factor of 6. Write 2 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 1 is 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 least common multiple, the division method is the most common and easy method that we use. Use the online LCM calculator to verify your answers.

LCM formulas are the collection of the numbers, their LCM, and their HCF (Highest Common Factor). 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)/HCF(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

The HCF or the highest common factor of two or more numbers is the highest or the greatest factor among all the common factors of the given numbers, whereas the LCM or the least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers. The following table shows the difference between HCF and LCM:

The least common multiple of two or more numbers is having a wide number of properties. Given below are the three most common properties of the LCM:

• Associative Property: This property is applicable for two numbers: LCM (a,b) = LCM (b,a).
• Commutative Property: This property is only applicable for three numbers: LCM (a,b,c) = LCM (a, LCM (b,c)) = LCM (LCM (a,b),c) = LCM (LCM (a,c),b).
• Distributive property: This property is only applicable for four numbers: LCM (da, db, dc) = d × LCM (a, b, c).

LCM of Numbers

• LCM of 10 and 12
• LCM of 10 and 15
• LCM of 10 and 20
• LCM of 11 and 12
• LCM of 11 and 15
• LCM of 12 and 13
• LCM od 12 and 14
• LCM of 12 and 15
• LCM of 12 and 16
• LCM of 12 and 18
• LCM of 12 and 20

What does Least Common Multiple Mean?

The least common multiple of two or more numbers is the lowest number that is a common multiple of the given numbers. If X and Y are two numbers such that X is a multiple of Y, then the LCM (X, Y) = X. LCM cannot be less than any of the given numbers except for 0 which can be considered as a common multiple of all whole numbers.

How to Find the Least Common Multiple?

There are 3 methods to calculate the LCM of two numbers: by listing out the common multiples, by prime factorization method, and by division method. (These methods have been explained in the above sections with examples)

What is the Fastest and Easy Way to Find the LCM?

Though we have three methods to find the LCM: listing out the common multiples, prime factorization method, and division method. Out of these three methods, the division method is the most common, fastest, and easiest method. It can be used for small or large numbers conveniently.

What is the LCM of 12 and 9?

First few multiples of 12 are 12, 24, 36, 48, 60, 72, 84, ... and first few multiples of 9 are 9, 18, 27, 36, 45, … The LCM of 12 and 9 is the smallest multiple that is common to both the numbers. Thus, the LCM of 12 and 9 is 36.

What is the Difference Between LCM and HCF?

The least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers and the HCF (Highest Common Factor) of two or more numbers is the highest number among all the common factors of the given numbers.

What is the Relationship Between HCF and LCM of Two 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.

What are the Properties of LCM?

Few important properties of LCM are given below:

• LCM of two relatively primes is always the product of those numbers.
• LCM of two or more prime numbers is always the product of those numbers.
• Associative Property: This property is applicable for two numbers: LCM (a,b) = LCM (b,a).
• Commutative Property: This property is only applicable for three numbers: LCM (a,b,c) = LCM (a, LCM (b,c)) = LCM (LCM (a,b),c).
• Distributive property: This property is only applicable for four numbers: LCM (da, db, dc) = d × LCM (a, b, c).

What is the Least Common Multiple of 8 and 12?

The LCM of 12 and 8 is the smallest number among all common multiples of both of the numbers. Multiples of 8 are 8, 16, 24, 32, 40, ... and the multiples of 12 are 12, 24, 36, ... Thus, the LCM of 12 and 8 is 24.