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. 

Common Multiples Definition

Common multiples are numbers that two or more given numbers share as multiples. In other words, a common multiple is a number that can be exactly divided by each of the given numbers. For example, 24 is a common multiple of 6 and 8 because both 6 and 8 divide 24 without leaving a remainder. 
 

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. 

The following table highlights the key differences between factors and multiples:
 

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. 

   \( \begin{aligned} 4 \times 1 &= 4 \\ 4 \times 2 &= 8 \\ 4 \times 3 &= 12 \\ 4 \times 4 &= 16 \\ 4 \times 5 &= 20 \\ 4 \times 6 &= 24 \end{aligned} \)

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

To find the multiples of 5, multiply 5 with each number: 
 

\( \begin{aligned} 5 \times 1 &= 5 \\ 5 \times 2 &= 10 \\ 5 \times 3 &= 15 \\ 5 \times 4 &= 20 \\ 5\times 5 &= 25\\ 5 \times 6 &= 30\end{aligned} \)

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 common multiples within the first 10 multiples of each are 12.

Note: 24 is also a common multiple but appears later in the sequence.

This is the only common multiple of 2, 3, and 4 in the first 10 multiples of each number.

Least Common Multiple (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 prime factorization, break each number into primes and multiply the highest powers of all primes present. 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. 
 

Given below are a few tips and tricks that can be used to make working with common multiples easier:

• Remember that LCM is the smallest number that all numbers can divide evenly.
   
•  Break each number into prime factors, then take the highest power of each prime numbers that appears. multiplying them together gives the LCM.
   
• For large numbers, use the formula \( \text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)} \)
   
• Spot patterns in multiplication tables by writing them side by side to visually spot common values.
   
• Remember that any number is a multiple of itself and multiples increase by equal intervals.
   
• Encourage children to use real life examples, like finding common multiples when setting equal groups of objects. 
   
• Parents and teachers can make use of visual aids such as colored number charts or multiplication tables to help students memorize the patterns of the common multiples.
   
• Provide simple practice tasks, like comparing multiples of small numbers during daily activities, to students. 
   
• Teachers can give group activities to students, where they can list multiples and identify common ones together. 
   
• Parents and teachers can provide students with digital tools, apps or common multiples worksheets where they can practice with multiplication tables to improve accuracy and speed. 
   

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. 
 

• Timing of traffic lights: Traffic lights at different intersections change after fixed intervals. LCM ensures smooth coordination.
   
• 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.
   
• Agricultural practices: Crop rotation is the practice of planting different crops in succession in order to sustain soil health and maximize nutrients in the soil. By planting crops that have different growing seasons, using a common multiple of their cycles, farmers can utilize their land efficiently across growing seasons.
   
• Astronomy: Predicting planetary alignments and eclipses relies on knowing the common multiples of the lengths of their orbital periods. This way, astronomers can predict when two or more bodies will again align in the night sky.