The factors of $12$ are $1, 2, 3, 4, 6$ and $12$.

The number $12$ is a natural number and it represents a whole quantity.

Now, let’s learn how to find the factors of twelve mathematically by factoring the number $12$ and it proves why $1, 2, 3, 4, 6$ and $12$ are factors of $12$.

Let’s divide the number $12$ by $1$ firstly.

$12 \div 1$

$=\,\,$ $\dfrac{12}{1}$

$=\,\,$ $12$

The number $1$ divides $12$ completely. So, the number $1$ is a factor of $12$.

Now, let’s divide the number $12$ by $2$.

$12 \div 2$

$=\,\,$ $\dfrac{12}{2}$

$=\,\,$ $6$

The number $2$ divides $12$ completely. Therefore, the number $2$ is a factor of $12$.

Similarly, let’s divide the number $12$ by $3$.

$12 \div 3$

$=\,\,$ $\dfrac{12}{3}$

$=\,\,$ $4$

The number $3$ divides $12$ completely. So, the number $3$ is a factor of $12$.

In the same way, let’s divide the number $12$ by $4$.

$12 \div 4$

$=\,\,$ $\dfrac{12}{4}$

$=\,\,$ $3$

The number $4$ divides $12$ completely. Therefore, the number $4$ is a factor of $12$.

Similarly, let’s divide the number $12$ by $5$.

$12 \div 5$

$=\,\,$ $\dfrac{12}{5}$

Let’s observe the division process of natural number $12$ by $5$.

$\require{enclose}

\begin{array}{rll}

2 && \hbox{} \\[-3pt]

5 \enclose{longdiv}{12}\kern-.2ex \\[-3pt]

\underline{-~~~10} && \longrightarrow && \hbox{$5 \times 2 = 10$} \\[-3pt]

\phantom{00} 2 && \longrightarrow && \hbox{Remainder}

\end{array}$

There is a remainder in the division and it expresses that the number $5$ cannot divide $12$ completely. So, the number $5$ is not a factor of $12$.

Now, let’s divide the number $12$ by $6$.

$12 \div 6$

$=\,\,$ $\dfrac{12}{6}$

$=\,\,$ $2$

The number $6$ divides $12$ completely. Therefore, the number $6$ is a factor of $12$.

Let’s try to divide the number $12$ by the numbers $7, 8, 9, 10$ and $11$.

$(1).\,\,$ $12 \div 7$ $\,=\,$ $\dfrac{12}{7}$ $\,=\,$ $1.7142\cdots$

$(2).\,\,$ $12 \div 8$ $\,=\,$ $\dfrac{12}{8}$ $\,=\,$ $1.5$

$(3).\,\,$ $12 \div 9$ $\,=\,$ $\dfrac{12}{9}$ $\,=\,$ $1.3333\cdots$

$(4).\,\,$ $12 \div 10$ $\,=\,$ $\dfrac{12}{10}$ $\,=\,$ $1.2$

$(5).\,\,$ $12 \div 11$ $\,=\,$ $\dfrac{12}{11}$ $\,=\,$ $1.0909\cdots$

The numbers $7, 8, 9, 10$ and $11$ are failed to divide $12$ completely. So, the numbers from $7$ to $11$ are not factors of $12$.

Finally, let’s divide the number $12$ by the same number.

$12 \div 12$

$=\,\,$ $\dfrac{12}{12}$

$=\,\,$ $1$

The factorization of $12$ is proved that the whole numbers $1, 2, 3, 4, 6$ and $12$ divided the number $12$ completely. So, the numbers $1, 2, 3, 4, 6$ and $12$ are called the factors of $12$.

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