Lowest Common Multiple, also known as LCM is the bedrock of arithmetic problems as it is used to perform basic arithmetic operations on fractions and some more complex arithmetic problems. In this article, we will discuss LCM, but first, we have to know what multiples are all about.

The multiples of a given number in Mathematics can be defined as all the numbers that are the results of multiplying the number and any other number. It is any set of numbers gotten from multiplying the number by an integer. For example, taking three numbers 2, 4 and 8;

The multiples of 2 are 2, 4, 6, 8, 10, 12…

The multiples of 4 are 4, 8, 12, 16, 20…

The multiples of 8 are 8, 16, 24, 32, 40…

LOWEST COMMON FACTOR

A factor in Mathematics can be defined as any number that can be multiplied to get a certain number. It can also be defined as a number that divides a certain number without leaving a remainder.

For example, considering the number 6, 1 can be multiplied by 6 and 2 can be multiplied by 3 to get 6, which means the factors of 6 are 1, 2, 3 and 6 as they are the only numbers that can divide 6 without leaving a remainder. Now, let’s focus on what lowest common factors are.

Lowest Common Multiple in Mathematics of a given set of numbers is also known as the lowest common factor of those numbers. The Lowest Common Multiple or LCM of two or more natural numbers is the lowest multiple the given numbers have in common. 

For example, consider the numbers 3, 6 and 15.

The factors of 3 are 1 and 3.

The factors of 6 are 1, 2 and 3.

The factors of 15 are 1, 3 and 5.

The common factors in the three given numbers are 1 and 3. To get the Lowest Common Factor, we multiply the common factors of all the numbers.

Therefore the LCM of 3, 6 and 15 = 1 x 2 x 3 x 5 = 30

Understanding the concept of Lowest common multiples or LCM plays an important part in the addition, subtraction and comparison of two or more fractions. To understand this better, we will talk about the lowest common denominator and how it is used in mathematical operations on fractions.

LOWEST COMMON DENOMINATOR

The lowest common denominator is the lowest common factor of the denominators of a set of fractions. The top layer of the fraction is known as the numerator while the lower layer is called the denominator. For example, considering the numbers 3/4 and 1/6; 3 and 1 are the numerators while 4 and 6 are the denominators.

To get the lowest common denominator, we will be dealing with the multiples of the denominators alone.

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36…

Multiples of 6 = 6, 12, 18, 24, 30, 36…

From the multiples of both numbers, 12 is the lowest common denominator.
    

LOWEST COMMON MULTIPLES EXAMPLES

There are three common ways to solve LCM.

1. Listing Method
2. Prime Factorization
3. Division Method
   

• LISTING METHOD

In this method, the multiples of the given set of numbers are listed and then the least multiple common to the set of numbers is noted as the LCM. Here are a few exercises to illustrate finding the Lowest Common Factor by the listing method.

1. Find the LCM of 2, 3 and 4.

Solution

The multiples of 2, 3 and 4 are:

Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24…

Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24…

Multiples of 4 = 4, 8, 12, 16, 20, 24…

The common multiples of 2, 3 and 4 are 12 and 24. 

Since 12 is the smallest common multiple, the Lowest Common Factor or LCM is 12.

2. Find the LCM of 5 and 6

Solution

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40…
             
Multiples of 6 = 6, 12, 18, 24, 30, 36, 42…

The common multiple between the two numbers is 30.

Therefore 30 is the lowest common multiple, the LCM is 30. That means 5 and 6 can divide 30 with no remainder.

3. Find the LCM of 8 and 10

Solution

Multiples of 8 = 8, 16, 24, 32, 40…

Multiples of 10 = 10, 20, 30, 40…

Therefore, the Lowest Common Multiple is 40.

• PRIME FACTORIZATION METHOD

Prime factors are the set of prime numbers that can be multiplied to give a certain number.  As shown in the Lowest Common Factor section example, all the factors listed there are prime factors. Here are some exercises that use prime factorization to find the Lowest Common Multiple of given numbers.

1. Find the LCM of 8, 18 and 24
   
   Solution.
   
   The prime factors of the given numbers respectively are:
   
   8 = 2 x 2 x 2
   
   18 = 2 x 3 x 3
   
   24 = 2 x 2 x 2 x 3
   
   Writing all the prime factors and sorting the common prime factors we have:
   
   8, 18 and 24 = 2 x 2 x 2 x 2 x 3 x 3 x 2 x 2 x 2 x 3
   
   To calculate the LCM we sort the prime factors common to the numbers and take the highest power of each;
   
   Therefore, the LCM = 2 x 2 x 2 x 3 x 3 = 72
   
2. Find the LCM of 8 and 10
   
   Solution.
   
   The prime factors of the given numbers are:
   
   8 = 2x2x2
   
   10 = 2 x5
   
   Therefore, the LCM = 2 x 2 x 2 x 5 = 8 x 5 = 40

3. Find the LCM of the following numbers: 12, 90 and 20.
   
   Solution.
   
   The prime factors of the given numbers are:
   
   12 = 2 x 2 x 3
   
   90 = 2 x 3 x 3 x 5
   
   20 = 2 x 2 x 5 

The LCM = 2 x 2 x 3 x 3 x 5 = 4 x 9 x 5 = 180

• DIVISION METHOD

In the division method of calculating LCM, the numbers are written in a table and then divided by the smallest prime factor of any of the given numbers written on the first column of the table. Any number that is not divisible by the prime factor is written down again after which the numbers are divided by more prime factors till the numbers are all 1 at the last row of the table.

The LCM is got by multiplying all the prime factors when the numbers are all equal to 1.  Here are a few examples that use the division method to find the Lowest Common Multiples of numbers.

1. Find the LCM of 12, 20 and 25 using the division method.

Solution

LCM = 2 x 2 x 3 x 5 x 5 = 300

2. Find the Least Common Factor of 30, 27, 60  and 84

Solution

LCM = 2 x 2 x 3 x 3 x 3 x 5 x 7 = 3780

Therefore the Lowest Common Factor of 30, 27, 60 and 84 is 3780.

KEY POINTS

1. LCM stands for Lowest Common Multiple.

2. Multiples are all the numbers that are the results of multiplying the number and an integer.

3. LCM is the same as the Lowest Common Factor (LCF).

4. There are three common ways to solve LCM which are the Listing Method, the Prime Factorization Method and the Division Method.

FAQ

1.  How do you find LCM?

The Lowest Common Multiple of a number can be found using different methods like listing, prime factorization or division method.

Quiz

1. What does LCM stand for?

Answer: LCM stands for Least or Lowest Common Multiple.

2. Which of the following is the LCM of 2, 3 and 4?

1. 6
2. 4
3. 2
4. 12

Answer: D

3. Which of these is not a method of finding LCM?

1. Division Method
2. Multiplication Method
3. Prime Factorization
4. Listing Method

      Answer = B