Factors and Multiples (With Methods & Examples)

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Factors and multiples are two related terms. If a number $a$ is a factor of another number $b$, then the number $b$ is a multiple of the first number $a$. 

A factor of a number is the number that divides it completely without leaving any remainder, whereas a multiple is a number that comes in the multiplication table of a number. For example, $2$ is a factor of $4$, as $2$ divides $4$ completely without leaving any remainder. Similarly, $4$ is a multiple of $2$ as $4$ comes in the multiplication table of $2$.

Factors and Multiples are Related to Multiplication and Division

Both factors and multiples are related to the arithmetic operations of multiplication and division.

If a number $a$ can be represented as $b \times c$, i.e., $a = b \times c$, then $b$ and $c$ are called factors of $a$ and $a$ is called as multiple of $b$ as well as $c$.

For example, $12 = 3 \times 4$, then $3$ and $4$ are factors of $12$ and $12$ is a multiple of $3$ as well as $4$.

Let’s take another example, $72 = 12 \times 6$, then here also, $12$ and $6$ are factors of $72$ and $72$ is a multiple of $12$ as well as $6$.

If a number $a$ can be represented as $b \div c$, i.e., $a = b \div c$, then $a$ and $c$ are called factors of $b$ and $b$ is called multiple of both $a$ and $c$.

For example, $5 = 15 \div 3$, then $5$ and $3$ are factors of $15$ and $15$ is a multiple of both $5$ and $3$.

Let’s consider another example, $8 = 56 \div 7$, then $8$ and $7$ are factors of $56$ and $56$ is called multiple of both $8$ and $7$.

How To Find Factors and Multiples of a Number?

As mentioned above the terms factors and multiples are related to the processes of multiplication and division. So, you can use the concept of multiplication and division to find the factors and multiples of any given number.

Finding Factors of a Number

Let’s consider some examples to understand the process of finding the factors of a number.

Ex 1: Find factors of $24$ 

Start from the number $1$ and find a corresponding number that on multiplying gives $24$. The number is $24$.

$1 \times 24 = 24$

Now, move on to the next number, i.e., $2$. Again find a corresponding number that on multiplying gives $24$. The number is $12$.

$2 \times 12 = 24$

Repeat the process for the next number, i.e., $3$.

$3 \times 8 = 24$

Next number: $4$.

$4 \times 6 = 24$

Next number: $5$. Notice that $24$ is not divisible by $5$, so move on to the next number, i.e., $6$.

$6 \times 4 = 24$

Notice that  $4 \times 6 = 24$ and $6 \times 4 = 24$ are same (Commutative Property of Numbers), so stop here and collect all the numbers that on multiplying give $24$. These are

$1 \times 24 = 24$

$2 \times 12 = 24$

$3 \times 8 = 24$

$4 \times 6 = 24$

Thus, the factors of $24$ are $1$, $2$, $3$, $4$, $6$, $8$, $12$ and $24$.

Note: 

• $1$ is a factor of every number
• The number itself is a factor of itself $\left(24 \text{ is a factor of 24} \right)$.

Ex 2: Find factors of $72$ 

Start from the number $1$ and find a corresponding number that on multiplying gives $72$. The number is $72$.

$1 \times 72 = 72$

Now, move on to the next number, i.e., $2$. Again find a corresponding number that on multiplying gives $72$. The number is $36$.

$2 \times 36 = 72$

Repeat the process for the next number, i.e., $3$.

$3 \times 24 = 72$

Next number:  $4$.

$4 \times 18 = 72$

The next number is $5$. But $72$ is not divisible by $5$, so move on to the next number, i.e., $6$.

$6 \times 12 = 72$

For the next number $7$, the number $72$ is not divisible, so move on to the next number $8$.

$8 \times 9 = 72$

Next number is $9$ and $9 \times 8 = 72$. Since the numbers start repeating, so stop the process and collect the number pairs.

$1 \times 72 = 72$

$2 \times 36 = 72$

$3 \times 24 = 72$

$4 \times 18 = 72$

$6 \times 12 = 72$

$8 \times 9 = 72$

Thus, the factors of $72$ are $1$, $2$, $3$, $4$, $6$, $8$, $9$, $12$, $18$, $24$, $36$ and $72$.

Note: The number itself is a factor of itself $\left(72 \text{ is a factor of 72} \right)$.

Finding Multiples of a Number

Multiples of a number are the numbers that we get after multiplying the number by a whole number (The numbers that you get in a multiplication table). 

To find the multiples of a number, note down all the numbers that result from the multiplication of a number by another whole number.

The skip counting method is one of the simplest methods to find the multiples of any given number.

Let’s consider some examples to understand the process of finding the multiples of a number.

Ex 1: Find the first few multiples of $17$.

Note: For any number, there are infinite (countless) multiples.

Multiples of $17$	Description
$17$	$17 \times 1$
$34$	$17 \times 2$, or $17 + 17$
$51$	$17 \times 3$, or $17 + 17 + 17$
$68$	$17 \times 4$, or $17 + 17 + 17 + 17$
$85$	$17 \times 5$, or $17 + 17 + 17 + 17 + 17$
$102$	$17 \times 6$, or $17 + 17 + 17 + 17 + 17 + 17$
$119$	$17 \times 7$, or $17 + 17 + 17 + 17 + 17 + 17 + 17$

First $7$ multiples of $17$ are $17$, $34$, $51$, $68$, $85$, $102$ and $119$.

Note: A number is a multiple of itself.

Ex 2: Find the first few multiples of $29$.

Multiples of $29$	Description
$29$	$29 \times 1$
$58$	$29 \times 2$, or $29 + 29$
$87$	$29 \times 3$, or $29 + 29 + 29$
$116$	$29 \times 4$, or $29 + 29 + 29 + 29$
$145$	$29 \times 5$, or $29 + 29 + 29 + 29 + 29$
$174$	$29 \times 6$, or $29 + 29 + 29 + 29 + 29 + 29$
$203$	$29 \times 7$, or $29 + 29 + 29 + 29 + 29 + 29 + 29$

First $7$ multiples of $29$ are $29$, $58$, $87$, $116$, $145$, $174$ and $203$.

Properties of Factors and Multiples

There are some properties of factors and multiples that are listed below.

• Factors and multiples are only used and applicable to whole numbers
• $1$ is the factor of every number
• For every number, $1$ is the smallest factor
• For every number, the number itself is the largest factor
• There is a finite number of factors for every number
• If there are only two factors of a number, i.e $1$ and the number itself, that number is called a prime number
• Factors of a number are always less than or equal to the given number 
• $0$ is the multiple of every number
• Every number is a multiple of itself
• There are infinite(or countless) multiples of every number
• Multiples of a number are always equal to or greater than the given number (except $0$)

Conclusion

The factors are multiples are related to each other. If $a$ is a factor of $b$, then $b$ is a multiple of $a$. For any given number there is a finite number of factors but the number of multiples is infinite.

Practice Problems

1. State True or False
   • $0$ is a factor of every number
   • $0$ is a multiple of every number
   • $1$ is a factor of every number
   • $1$ is a multiple of every number
   • Every number is a factor of itself
   • Every number is a multiple of itself
   • For every number, there is a finite number of factors
   • For every number, there is a finite number of multiples
   • For every number, there is an infinite number of factors
   • For every number, there is an infinite number of multiples
2. Find all the factors of the following numbers
   • $128$
   • $56$
   • $98$
   • $78$
   • $132$
3. Find first $8$ multiples of the following numbers
   • $13$
   • $19$
   • $21$
   • $32$
   • $64$

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties
• Rationalize The Denominator(With Examples)
• Multiplication of Irrational Numbers(With Examples)
• Rationalize The Denominator(With Examples)
• Division of Irrational Numbers(With Examples)

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