Multiples: A multiple is a result of multiplying one integer by another. Multiples are just the product of two elements. To discover multiples of \(2\), for example, multiply \(2\) by \(1,2\) by \(2,2\) by \(3,2\) by \(4\), and so on. These numbers are multiplied together to form the multiples.

Multiples and factors are important concepts to learn when studying mathematics. In this post, we will define multiples and show you how to determine the multiples of a number. We’ll also go through common multiples and their examples. Scroll down to find more.

Define Multiples

We can get the multiple by multiplying a number by a positive integer. We can get the multiples of the whole numbers by taking out the product of the counting numbers and that of whole numbers. Let’s say we have a number \(x\) and the counting numbers as \(1,2,3,4,5\) Then, the product of \(x\) and \(1,2,3,4,5 \ldots\). 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.

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Let us find the multiples of \(3\).

\(3 \times 1=3,3 \times 2=6,3 \times 3=9,3 \times 4=12\) and so on.

Thus, the multiples of \(3\) are \(3,6,9,12,15,18,21,24,27,30,33,36, \ldots\)

List of Multiples

We can list out the multiples of a number by multiplying the given number to another number. For example, the below-given list shows the first \(10\) multiples of some numbers.

Number	Multiples
\(4\)	\(4, 8, 12, 16, 20, 24, 28, 32, 36, 40\)
\(5\)	\(5, 10, 15, 20, 25, 30 ,35, 40, 45, 50\)
\(6\)	\(6, 12, 18, 24, 30, 36, 42, 48, 54, 60\)
\(9\)	\(9, 18, 27, 36, 45, 54, 63, 72, 81, 90\)
\(10\)	\(10, 20, 30, 40, 50, 60, 70, 80, 90, 100\)
\(12\)	\(12, 24, 36, 48, 60, 72, 84, 96, 108, 120\)
\(15\)	\(15, 30, 45, 60, 75, 90, 105, 120, 135, 150\)
\(16\)	\(16, 32, 48, 64, 80, 96, 112, 128, 144, 160\)
\(19\)	\(19, 38, 57, 76, 95, 114, 133, 152, 171, 190\)

Properties of Multiples

1. Every number is a multiple of \(1\).
For example, \(7 \times 1=7,20 \times 1=20,55 \times 1=55\)
2. Every number is a multiple of itself.
For example, \(1 \times 7=7,1 \times 19=19,1 \times 100=100\)
3. Every multiple of a number is either equal to or greater than the number.
For example, the multiples of \(7\) are \(7,14,21,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126 \ldots\)
4. The smallest multiple of a number is the number itself.
5. The multiples of an even number are always even. For example, the multiples of \(2\) are \(2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40\)
6. The multiples of an odd number are alternatively odd and even. For example, the multiples of \(3\) are \(3,6,9,12,15,18,21,24,27,30,33,36,39,42, \ldots\)
7. There is no end to multiples of a number. For example, the multiples of \(5\) are \(5,10,15,20,25,30, \ldots \ldots, 105,110, \ldots \ldots 1000,1005, \ldots \ldots\)

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Definition of Common Multiples

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.

Let us understand the concept of common multiples with the help of an example.

Consider two numbers, \(15\) and \(20\).

Multiples of \(15\) and \(20\) are,

\(15=15,30,45,60,75,90,105,120,135,150,165,180,195,210,225,240 \ldots \ldots\)
\(20=20,40,60,80,100,120,140,160,180,200,220,240,260,280, \ldots \ldots\)

We see that \(60\) and \(120\) are the first two common multiples of \(15\) and \(20\).

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Real-Life Applications of Common Multiples

The concept of common multiples can be applied in real life as well. Want to know how common multiples are helpful in our daily life? 

Let us consider the example:

Priya can complete one lap in \(8\) minutes, and Preeti can complete it in \(6\) minutes. How long will it take for both to arrive at the same starting point together, if they start at the same time and maintain their running pace? 

To find the time needed for both to arrive at the same starting point together can be solved by finding the common multiples of \(8\) and \(6\).

The multiples of \(8=8,16,24,32,40,48,56,64,72,80, \ldots\).
The multiples of \(6=6,12,18,24,30,36,42,48,54,60,66, \ldots\).
The common multiples of \(8\) and \(6\) are \(24,48, \ldots\)

We can see that \(24\) is the first common multiple.

Thus, Priya and Preeti will meet again after \(24\) minutes.

Lowest Common Multiple

LCM stands for lowest common multiples. The LCM of two or more given number is the lowest or smallest number that is a multiple of each of the given numbers. 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 \(3=3,6,9,12,15,18,21,24,27,30,33,36,39,42, \ldots\)
The multiples of \(5=5,10,15,20,25,30,35,40,45,50,55, \ldots\)
Common multiples of \(3\) and \(5=15,30,45, \ldots\)

The lowest common multiple of \(3\) and \(5\) is \(15\).

If checked carefully, you will find that \(15\) is the smallest number that is exactly divisible by \(3\) and \(15\).

Methods of Finding Lowest Common Multiple

The following three methods are most commonly used to find the LCM. 

Common Multiple Method

Let us understand the concept of common multiple methods with the help of an example.

For example: Find the common multiples of \(4\) and \(5\).

Multiples of \(4=4,8,12,16,20,24,28,32,36,40, \ldots \ldots\)
Multiples of \(5=5,10,15,20,25,30,35,40,45,50, \ldots \ldots\)
Now, common multiples of \(4\) and \(5\) are \(20,40, \ldots\)

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

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Prime Factor Method

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

Example: Find the LCM of \(12\) and \(15\).

Prime factorization of \(12=2 \times 2 \times 3\)
Prime factorization of \(15=3 \times 5\)
Product of the common prime factors of \(12\) and \(15=3=\mathrm{HCF}\)
Product of the remaining factors \(=2 \times 2 \times 5\)
The LCM \(=\) HCF \( \times \) product of the remaining prime factors

Thus, the LCM \(=3 \times 2 \times 2 \times 5=60\)

Hence, the LCM of \(12\) and \(15=60\)

Common Division Method

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

Find the LCM of \(16, 20\) and \(24\).

Write all the numbers in a horizontal line, separating them by commas.
Divide by a suitable number that exactly divides at least two of the given numbers.
And write down the quotients and the undivided numbers obtained below the first line. 
Repeat the process till we get a line of numbers that are prime to each other.

LCM will be the product of the quotients and the prime factors in the last row.

Therefore, LCM \(=2 \times 2 \times 2 \times 2 \times 3 \times 5=240\)

Solved Examples – Multiples

Q.1. Find the multiples of \(1\).
Ans: The multiples of \(1=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, \ldots\).

Q.2. Find the common multiples of \(15\) and \(25\) .
Ans: Multiples of \(15=15,30,45,60,75,90,105,120,135,150,165,180,195,210\),
Multiples of \(25=25,50,75,100,125,150,175,200,225,250, \ldots\)
Common multiples of \(15\) and \(25=75,150, \ldots\)

Q.3. Find the multiples of the given following numbers.
a) \(14\) b) \(21\) c) \(23\) d) \(50\)
Ans: The multiples of the given numbers are as follows:
(a) The multiples of \(14=14,28,42,56,70,84,98,112,126,140,154,168, \ldots\)
(b) The multiples of \(21=21,42,63,84,105,126,147,168,189,210, \ldots\)
(c) The multiples of \(23=23,46,69,92,115,138,161,184,207,230, \ldots\)
(d) The multiples of \(50=50,00,150,200,250,300,350,400,450,500, \ldots\)

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

Q.5. Find the multiples of \(11\) and \(33\).
Ans: The multiples of \(11=11,22,33,44,55,66,77,88,99, \ldots\)
The multiples of \(33=33,66,99,132,165,198,231,264, \ldots\)

Q.6. Find the smallest number which, when divided by \(8,12,16,24\) and \(36\), leaves no remainder.
Ans: The smallest number that is exactly divisible by each given number is their lowest common multiple.
Therefore, the required number \(=\) LCM of \(8,12,16,24\) and \(36\)

Hence, the LCM \(=2 \times 2 \times 2 \times 3 \times 2 \times 3=144\)
Thus, the smallest number that is exactly divisible by \(8,12,16,24\) and \(36\) is \(144\).

Q.7. Find the smallest number which, when:
a) Increased by \(3\)                 b) decreased by \(1\)
is exactly divisible by \(21, 45, 63, 81\) and \(210\).
Ans: First, find the LCM of the given numbers.

Therefore, the LCM\(=3×3×5×7×3×3×2=567\)
a) On increasing \(5667\) by \(3\) , we get \(5667+3=5670\), which is the LCM of the given
numbers, i.e., exactly divisible by each of them.
Hence, the required number \(=5670-3=5667\)

b) On decreasing \(5671\) by \(1\), we get \(5671-1=5670\), which is the LCM of the given
numbers, i.e., exactly divisible by each of them.
Hence, the required number \(=5670+1=5671\)

Summary

In this article, we learned the definitions of multiples, common multiples and least common multiples and also understood the concept of finding LCM. In addition to this, we learned the properties of multiples and understood the concept by applying it in real-life situations.

Frequently Asked Questions (FAQ) – Multiples

Q.1. What are \(4\) multiples of any \(4\) numbers?
Ans: The multiples of any \(4\) numbers are as follows:

\(7\)	\(7, 14, 21, 28\)
\(11\)	\(11, 22, 33,44\)
\(20\)	\(20, 40, 60, 80\)
\(40\)	\(40, 80, 120, 160\)

Q.2. What are the multiples of \(3\) ?
Ans: The multiples of \(3\) are \(3,6,9,12,15,18,21,24,27,30,33,26,39,42, \ldots\).

Q.3. Define multiples with the help of examples.
Ans: We can get the multiple by multiplying a number by a positive integer. We can get the multiples of the whole number by taking out the product of the counting numbers and that of whole numbers. For example, the multiples of \(3\) and \(5\).
The multiples of \(3=3,6,9,12,15,18,21,24,27,30,33,36,39,42, \ldots\)
The multiples of \(5=5,10,15,20,25,30,35,40,45,50,55, \ldots\)

Q.4. What are the multiples of \(13\) ?
Ans: The multiples of \(13\) are \(13,26,39,52,65,78,91,104,117,130,143, \ldots\).

Q.5. What are common multiples? Explain with the help of an example.
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.
Consider two numbers, \(15\) and \(40\).
Multiples of \(15\) and \(40\) are,
\(15=15,30,45,60,75,90,105,120,135,150,165,180,195,210,225,240 \ldots \ldots\)
\(40=40,80,120,160,200,240,280, \ldots \ldots\)
We see that \(120\) and \(240\) are the first two common multiples of \(15\) and \(40\).

Q.6. What are the multiples of \(12\) ?
Ans: The multiples of \(12\) are \(12,24,36,48,60,72,84,96,108,120,132,144,156, \ldots\).

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We hope this detailed article on 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.