Common Multiples

When we multiply a number by another number, the result that we get is called a multiple. For example, the multiples of 5 are 5, 10, 15, 20, 25, 30,...and so on. Here, the number 5 is multiplied by 1, 2, 3, 4, 5, 6, … and so on. The multiples of two or more numbers are called common multiples. 

For example, the multiples of 2 are 2, 4, 6, 8, 10, 12,..., and the multiples of 3 are 3, 6, 9, 12, 15, 18,...therefore, 6, 12, 18, 24, 30,…are the common multiples of 2 and 3. The smallest common multiple is called the Least Common Multiple (LCM), which in this case is 6. 
 

A multiple of a number is the product of that number and any whole number. For example, the multiples of 5 are 5, 10, 15, 20, 25, 30, etc. On the other hand, a factor is a whole number that divides another number evenly. For example, the factors of 5 are 1 and 5; 5 is a prime number and the factors of a prime number are always 1 and the number itself. 

Let’s understand this better using a table

The multiples of a given number can be found by multiplying the number with natural numbers like 1, 2, 3, 4, 5, and so on. For example, if you want to find the multiples of 4, multiply 4 by 1, 2, 3, 4, 5, 6, etc. 

    4 x 1 = 4
    
    4 x 2 = 8

    4 x 3 = 12

    4 x 4 = 16

    4 x 5 = 20

    4 x 6 = 24

So, the multiples of 4 are 4, 8, 12, 16, 20, 24,... and it goes on forever. 

We can also find the multiples of a number by repeatedly adding the number to itself. For example, to find the multiples of 10, we keep adding 10 each time. The multiples we get are 10, 20, 30, 40, etc.
 

To find the common multiples of two numbers, we first list the multiples of each number. Then, we look for the numbers that appear in both lists, called common multiples. For example, let’s take 3 and 4 and common multiples in the first 10 multiples. 

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40.

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

Just like a common multiple of two, to find the common multiples of three numbers, we first list the multiples of each number. Then, we identify the numbers that are present in all three lists. For example, let’s consider the first 10 multiples of 2, 3, and 4 and find the common multiples among them. 

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

The number that is common to all three numbers is 12. This is the only common multiple of 2, 3, and 4 in the first 10 multiples of each number.
 

LCM is the smallest multiple that can be divided evenly by two or more numbers. There are different methods to calculate LCM like the listing method, prime factorization, and division. For example, let’s find the LCM of 4 and 6:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28,...

Multiples of 6: 6, 12, 18, 24, 30, 36,...

The common multiples of 4 and 6 are 12, 24, 36, and so on. The smallest common multiple is 12. Therefore, the LCM of 4 and 6 is 12, as it is the smallest number that is divisible by both 4 and 6. 

Let’s look at the multiples of the first 10 numbers in a table. 

Common multiples play an important role in solving real-life problems where events or actions repeat at regular intervals. Here are some everyday situations where common multiples are used. 

• Traffic lights timing: Traffic lights at different intersections change after fixed intervals. To make them work smoothly, the time intervals are set using the Least Common Multiple 

• School Timetable: If a music class happens every 3 days and a sports class every 5 days, the LCM helps find when both classes will happen on the same day. 

• Event Planning: When planning events, LCM is used to find when two or more activities will occur together, like scheduling buses or cleaning schedules.