Common Multiple: Before we understand the concept of common multiples, first, let us know what a multiple is. A multiple is the product of a number by counting numbers and is exactly divisible by the given number. In simple words, multiples are the product of two factors. Now, come back to common multiples. Whenever we think about common multiples, immediately, more than one quantity comes to our mind. Thus, common multiples are multiples that two numbers have in common

In this article, we will cover and study common multiples, and by the time we reach the end of the article, we will expertise ourselves in finding the common multiples.

What is a Multiple?

A multiple is the product of a number by counting numbers and is exactly divisible by the given number. Let say we have a number \(x\) and the counting numbers as \(1,\,2,\,3,\,4,\,5…\)Then, the product of \(x\) and \(1,\,2,\,3,\,4,\,5…\) are called multiples, and this whole process or scenario is called multiplication.

For example, to find the multiples of \(5\), we multiply \(5\) by \(1\),\(5\) by \(2\),\(5\) by \(3\), and so on. Thus, the multiples are the product of this multiplication.

Common Multiples

The common multiples are the whole numbers that are shared multiples of numbers. When a number is a multiple of two or more numbers, it is called a common multiple of those numbers. Let us understand the concept of common multiples with the help of a couple of examples.

Consider two numbers \(2\) and \(3\)

Multiples of \(2\) and \(3\) are,
\(2 = 2,\,4,\,6,\,8,\,10,\,12,\,14,\,16,\,18,\,20,\,22,\,24,\,26,\,28,\,30,\,32,\,34,\,…..\)
\(3 = 3,\,\,6,\,9,\,12,\,15,\,18,\,21,\,\,24,\,27,\,\,30,\,33,\,\,…..\)

Common multiples \(2\) and \(3\) are \(6,\,\,12,\,\,18,\,\,24,\,30,……\)

Let us take another example with bigger numbers.

Consider two numbers, \(15\) and \(40\)

Multiples of \(15\) and \(40\) are,
\(15 = 15,30,45,60,75,90,\,105,\,120,\,135,\,150,\,165,\,180,\,195,\,210,\,225,\,240……..\)
\(40 = 40,\,80,\,120,\,160,\,\,200,\,\,240,\,280,\,……….\,\)

We see that \(120\) and \(240\) are the first two common multiples of \(15\) and \(40\) There can be many more common multiples between them.

But can there be the real-life use of common multiples? Absolutely, yes!!!

Suppose Manu and Gaurav are running on a circular track. They start from the same point, but Manu takes \(30\) seconds to cover a lap while Gaurav takes \(45\) seconds to cover the lap. So when will be the first time they meet again at the starting point?

This can be solved by finding the common multiples of \(30\) and \(45\).

The multiples of \(30\) and \(45\) are;
\(30 = 30,60,120,150,180,210,240,270…\,..\)
\(45 = 45,90,135,\,190,\,225,\,270…\,\,..\)

We can see that \(90\) is the first common multiple.

Thus, Manu and Gaurav will meet again after \(90\) seconds.

Common Multiple Examples

Example \(1\): Find the common multiples of \(3\) and \(5\)

The multiples of \(3 = 3,\,6,\,9,\,12,\,15,\,18,\,21,\,24,\,27,\,30,……\)
The multiples of \(5 = 5,\,10,\,15,\,20,\,25,\,30,……\)

So, the common multiples of \(3\) and \(5\) are \(15\),\(30,…..\)

Example \(2\): Find the common multiples of \(7\) and \(9\)

The multiples of \(7 = 7,\,14,\,21,\,28,\,35,\,42,\,49,\,56,\,63,\,70,\,77,\,84,\,91,98,\,105,\,112,\,119,126….\)
The multiples of \(9 = 9,18,27,\,36,\,45,\,54,\,63,\,72,\,81,90,99,\,108,\,117,126,….\)

So, the common multiples of \(7\) and \(9\) are \(63\), \(126,….\)

Lowest Common Multiple

The multiples common to two or more numbers are known as the common multiples. The smallest number among the common multiple is called the lowest common multiple or LCM. Thus, it is the smallest number that is exactly divisible by each of the given numbers.

Let us understand the concept of LCM with the help of an example.

Let us consider \(3\) and \(5\).
The multiples of \(3 = 3,\,\,6,\,\,9,\,12,\,15,\,18,\,\,21,\,\,24,\,\,27,30,\,33,\,\,36,\,39,\,42,….\)
The multiples of \(5 = 5,\,\,10,\,\,15,\,20,\,25,\,30,\,\,35,\,\,40,\,\,45,50,\,55,….\)

Common multiples of \(3\) and \(5\) \( = 15,\,30,\,45,\,….\)
Thus, the lowest common multiple among the common multiples is \(15\)

Hence, \({\rm{LCM}} = 15\)

Methods for Finding Lowest Common Multiple

The following three methods are most commonly used to find the

1. Common Multiple Method
2. Prime Factor Method
3. Common Division Method

Let us discuss and understand each method one by one.

Common Multiple Method

The following steps need to be followed to find the LCM by the common factor method. They are as follows;

1. Find the first few multiples of each given number.
2. From the multiples obtained in step \(1\) select the common ones.
3. The lowest number or multiple obtained in step \(2\) is the required lowest common multiple of the given numbers.

For example: Find the common multiples of \(4\),\(5\) and \(10\).
Multiples of \(4 = 4,\,\,8,\,\,12,\,\,16,\,\,20,\,\,24,\,\,28,\,\,32,\,\,36,\,\,40,……\)
Multiples of \(5 = 5,\,10,\,15,\,20,\,25,\,30,\,35,\,40,\,45,\,50,…..\)
Multiples of \(10 = 10,\,20,\,30,\,40,\,50,\,60,\,70,\,80,\,90,\,100,….\)

Now, common multiples of \(4\),\(5\) and \(10\) are \(20\),\(40,…..\)

Thus, LCM of \(4\),\(5\) and \(10\) is \(20\)

Prime Factor Method

Let us understand the concept of finding LCM with the prime factor method with the help of an example.

For example: Find the LCM of \(18,24\) and \(26\)

1. Express each of the given numbers as a product of its prime factors and then in the exponent or index form.

Clearly, \(18 = 2 \times 3 \times 3 = 2 \times {3^2}\)
\(24 = 2 \times 2 \times 2 \times 3 = {2^2} \times 3\)
\(36 = 2 \times 2 \times 2 \times 3 = {2^2} \times {3^2}\)

2. LCM is equal to all the prime factors obtained with the highest power. Since the prime factors \(2\) and \(3\), obtained above, with the highest power, are \({2^3}\) and \({3^2}\)

Therefore, the required \({\rm{LCM}} = {2^3} \times {3^2} = 2 \times 2 \times 2 \times 3 \times 3 = 72\)

Common Division Method

Let us understand the concept of finding LCM with the common division method with the help of an example.

For example: Find the LCM of \(6\), \(12\) and \(15\)

1. Start dividing with the smallest prime number, which can divide at least one of the given numbers. Next, bring down the number/ numbers which are not divisible as it is.
2. Continue this process till the last row has quotients for \(1\) all the given numbers.

3. Multiply all the prime factors to obtain the LCM.
\({\rm{LCM}} = 2 \times 2 \times 3 \times 5 = 60\)
Hence, the LCM of \(6\), \(12\) and \(15\)\( = 60\)

Solved Examples – Common Multiples

Q.1. Using the common multiple method, find the LCM of \(12\),\(15\) and \(20\)
Ans: Multiples of \(12 = 12,\,24,\,36,\,48,\,60,\,72,\,84,\,96,\,108,\,120,\,132…..\)
Multiples of \(15 = \,15,\,30,\,45,\,60,\,75,\,90,\,105,\,120,\,135,\,150,\,165,\,180,\,195,\,210,\,225,\,240,\,…\)
Multiples of \(20 = \,20,\,40,\,60,\,80,\,100,\,120,\,140,\,160,\,180,…..\)
Now, common multiples \(12,15\) and \(20\) are \(60,120,….\)
Thus, LCM of \(12,15\) and \(20\) is \(60\)

Q.2. Find the multiples of \(11\) and \(33\).
Ans: Multiples of \({\rm{11 = }}\,{\rm{11,}}\,{\rm{22,}}\,{\rm{33,}}\,{\rm{44,}}\,{\rm{55,}}\,{\rm{66,}}\,{\rm{77,}}\,{\rm{88,}}\,{\rm{99,}}\,……\)
Multiples of \({\rm{33}}\,{\rm{ = }}\,{\rm{33,}}\,{\rm{66,}}\,{\rm{99,}}\,{\rm{132,}}….\)
The common multiples of \(11\) and \(33\) are \({\rm{33,}}\,{\rm{99,}}….\)
The first common multiple or the least common multiple of both the numbers is \(33\) Hence, the \({\rm{LCM}}\,{\rm{ = }}\,{\rm{33}}{\rm{.}}\)

Q.3. Find the LCM of \(12\) and \(15\) by the prime factorisation method.
Ans: : Prime factors of \(12\, = \,2\, \times 2 \times 3\)
Prime factors of \({\rm{15}}\,{\rm{ = }}\,{\rm{3}}\,\, \times \,5\)
Therefore, LCM \({\rm{ = }}\,{\rm{2}} \times {\rm{2}} \times {\rm{3}} \times {\rm{5}}\,{\rm{ = }}\,{\rm{60}}\)
Hence, the LCM of \(12\) and \(15\) \({\rm{ = 60}}\)

Q.4. Find the common multiples of \(15\) and \(25\).
Ans: Multiples of \({\rm{15}}\,{\rm{ = }}\,15,\,30,\,45,\,60,\,75,\,90,\,105,\,120,\,135,\,150,\,165,\,180,\,195,\,210,….\)
Multiples of \(25\, = \,25,50,\,75,\,100,\,125,\,150,\,175,\,200,\,225,\,250,….\)
Common multiples of \(15\) and \(25\, = \,75,150\)

Q.5. Find the common multiples of \(2,3\) and \(4\).
Ans: Multiples of \(2 = \,2,\,4,\,6,\,8,\,10,\,12,\,14,\,16,\,18,\,20,\,22,\,24,\,26,\,28,\,30,\,32,\,34,\,36,\,38,\,40\)
Multiples of \(3 = 3,6,\,9,\,12,\,15,\,18,\,21,\,24,\,27,\,30,\,33,\,36,\,39,\,42,…..\,\)
Multiples of \(4 = \,4,\,8,\,12,\,16,\,20,\,24,\,28,\,32,\,36,\,40,\,…..\,\)
The common multiples of \(2,3\) and \(4\) are \(12,\,24,\,36,….\)

Summary

In this article, we learned about the multiples, common multiples and least common multiple. We also learned the application of common multiples in day-to-day life. We made ourselves aware that with the help of common multiples, we can find the least common multiple, and that’s why we emphasise more on the learning of different ways of finding the least common multiple.

Frequently Asked Questions (FAQ) – Common Multiples

Q.1. What are the common multiples of \(7\) and \(14\)?
Ans: The multiples of \(7\) and \(14\) are as follows
\(7 = \,7,\,14,\,21,\,28,\,35,\,42,\,49,\,56,\,63,\,70,\,77,\,84,\,91,\,98,\,105,\,112,\,119,\,126,\,…..\)
\(14\, = \,14,28,\,42,\,56,\,70,\,84,\,98,\,112,\,126,\,140,\,154,\,168,…..\)
So, the common multiples of \(7\) and \(14\) are \(14,\,28,\,42,\,56,\,70,….\)

Q.2. What is a common multiple of \(4\) and \(6\)?
Ans: The multiples of \(4\) and \(6\) are as follows,
\(4 = 4,\,8,\,12,\,16,\,20,\,24,\,28,\,32,\,36,\,40,….\)
\(6\, = \,6,\,12,\,18,\,24,\,30,36,\,42,\,….\)
So, the common multiple of \(4\) and \(6\) is \(12\).

Q.3. What are common multiples?
Ans: The common multiples are the whole numbers that are shared multiples of each set of numbers. Thus, the multiples common to two or more numbers are known as the common multiples of those numbers.
Consider two numbers: \(15\) and \(40\)
Multiples of \(15\) and \(40\) are,
\(15 = 15,\,30,\,45,\,60,\,75,\,90,\,105,\,120,\,135,150,\,165,\,180,195,\,210,\,225,240…\)
\(40\, = \,40,\,80,\,120,\,160,\,200,\,240,\,280,…\)
We see that \(120\) and \(240\) are the first two common multiples of \(15\) and \(40\)

Q.4. What are the common multiples of \(10\) and \(15\)?
Ans: The multiples of \(10\) and \(15\) are as follows
\(10\,\, = \,10,\,20,\,30,\,40,\,50,\,60,\,70,\,80,\,90,\,100,\,110,120,\,130,\,140,\,150,\,…..\)
\(15 = 15,\,30,\,45,\,60,\,75,\,90,\,105,120,\,135,\,150,165,\,180,\,195,\,210,\,225,\,….\)
So, the common multiple of \(10\)and\(15\) are \(3,\,60,\,90,\,120,\,150,\,180,\,……\,\)

Q.5. How do you find a common multiple of a number?
Ans: We can find the common multiple of two or more numbers by first listing the multiples of each number and then finding their common multiples. 

We hope this detailed article on common multiples has helped you in your studies. If you have any doubts or queries on this topic, you can comment down below and we will be more than happy to help you.