# Factorization of a Number

## Definition

The process of finding the factors of a number is called the factorization of a number.

### Introduction

Once you know the concept of a factor, you should learn how to find the factors of a number mathematically. There is a mathematical procedure to find the factors of a number in mathematics and it is called the factorization (U.S English) or factorisation (U.K English) of a number.

In mathematics, a number is divided by another number by the division method and the same division method is used in the factorization of a number. So, let’s learn the procedure of factoring a number to find the factors of a number mathematically. Three simple steps are used in the factorisation of a number.

1. Write the natural number from one up to a specific number.
2. Divide the number by the numbers from one up to the same number to find their corresponding quotients.
3. If the quotient is a natural number or an integer, then the number that divides a number is called a factor of that number. Otherwise, it is not a factor of that number.

The above three steps may confuse you theoretically, but you can easily understand here by the following simple and understandable example.

### Example

Let’s find the factors of a number $6$ to know how to find the factors of every number.

The first number in natural numbers is one. So, let’s write the numbers firstly from one to six in a row.

$1,$ $2,$ $3,$ $4,$ $5$ and $6$

Now, divide the number six by each of above numbers to find their corresponding quotient. The first number in natural numbers is one. So, let’s divide the six by one firstly.

#### Divide the number by One

The first natural number is one and let’s divide the number six by one to find their quotient.

$6 \div 1$

$\implies$ $6 \div 1 \,=\, \dfrac{6}{1}$

$\implies$ $\dfrac{\cancel{6}}{\cancel{1}} \,=\, 6$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{1} \,=\, 6$

It is evaluated that the quotient of $6$ divided by $1$ is equal to $6$ and it is a whole number, which means the number $1$ divides the number $6$ completely. So, the number $1$ is called a factor of $6$.

#### Divide the number by Two

The second natural number is two and let’s divide the number six by two to calculate their quotient.

$6 \div 2$

$\implies$ $6 \div 2 \,=\, \dfrac{6}{2}$

$\implies$ $\dfrac{\cancel{6}}{\cancel{2}} \,=\, 3$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{2} \,=\, 3$

It is calculated that the quotient of $6$ divided by $2$ is equal to $3$ and the number $3$ is a whole number, which means the number $2$ divides the number $6$ completely. Hence, the number $2$ is called a factor of $6$.

#### Divide the number by Three

The third natural number is three and let’s divide the number six by three to evaluate their quotient.

$6 \div 3$

$\implies$ $6 \div 3 \,=\, \dfrac{6}{3}$

$\implies$ $\dfrac{\cancel{6}}{\cancel{3}} \,=\, 2$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{3} \,=\, 2$

It is evaluated that the quotient of $6$ divided by $3$ is equal to $2$ and the number $2$ is a whole number, which means the number $3$ divides the number $6$ completely. Therefore, the number $3$ is called a factor of $6$.

#### Divide the number by Four

The fourth natural number is four and let’s divide the number six by four to find their quotient.

$6 \div 4$

$\implies$ $6 \div 4 \,=\, \dfrac{6}{4}$

$\implies$ $\dfrac{\cancel{6}}{\cancel{4}} \,=\, \dfrac{3}{2}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{4} \,=\, \dfrac{3}{2}$

It is calculated that the quotient of $6$ divided by $4$ is equal to $3$ over $2$, which is not a whole number and it is a rational number or a fraction. Similarly, the $3$ divided by $2$ is equal to $1.5$ in decimal form.

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{4} \,=\, 1.5$

It clears the number $4$ does not divide the number $6$ completely. So, the number $4$ is not a factor of $6$.

#### Divide the number by Five

The fifth natural number is five and let’s divide the number six by five to evaluate their quotient.

$6 \div 5$

$\implies$ $6 \div 5 \,=\, \dfrac{6}{5}$

It clears that the $6$ divided by $5$ is a fraction or a rational number and it is equal to $1.2$ in decimal number form.

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{5} \,=\, 1.2$

It clears the number $5$ does not divide the number $6$ completely. So, the number $5$ is not a factor of $6$.

#### Divide the number by Six

The sixth natural number is six and let’s divide the number six by itself to calculate their quotient.

$6 \div 6$

$\implies$ $6 \div 6 \,=\, \dfrac{6}{6}$

$\implies$ $\dfrac{\cancel{6}}{\cancel{6}} \,=\, 1$

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{6}{6} \,=\, 1$

The number $6$ divides $6$ completely. So, the number $6$ is called a factor of $6$.

The above steps have proved following two things.

1. The numbers $1,$ $2,$ $3$ and $6$ divide the number $6$ completely. So, they are called the factors of $6$.
2. The numbers $4$ and $5$ do not divide the number $6$ completely, So, they are not called the factors of $6$.

Thus, the factors of any number can be found easily by using above simple mathematical approach.

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