Factors of 10
Definition
A non-zero integer that divides $10$ exactly without leaving any remainder is called a factor of $10$.
What are the Factors of 10?
The factors of $10$ are $1$, $2$, $5$, and $10$.
Introduction to the Factors of 10
A factor of $10$ is an integer that divides $10$ exactly with no remainder. In arithmetic, integers include positive whole numbers, negative whole numbers, and zero, but zero cannot divide $10$. Therefore, zero is not considered when finding the factors of $10$, and all non-zero integers that divide $10$ exactly are called factors of $10$.
For every positive factor of $10$, there is a corresponding negative factor. Therefore, the complete list of factors of $10$ includes both positive and negative integers, as given below.
- The positive factors of $10$ are $1$, $2$, $5$, and $10$.
- The negative factors of $10$ are $-1$, $-2$, $-5$, and $-10$.
How to Find the Factors of 10
Let’s learn how to find the factors of $10$ by checking which numbers divide it exactly.
Proof
$10 \div 1$
Divide $10$ by $1$ to check whether any remainder is obtained.
$\require{enclose}
\begin{array}{rll}
10 && \hbox{} \\[-3pt]
1 \enclose{longdiv}{~10}\kern-.2ex \\[-3pt]
\underline{-~~~10} && \longrightarrow && \hbox{$1 \times 10 = 10$} \\[-3pt]
\phantom{00} 0 && \longrightarrow && \hbox{No Remainder}
\end{array}$
When $10$ is divided by $1$, there is no remainder. This means that $1$ divides $10$ exactly. Therefore, $1$ is a factor of $10$.
Proof
$10 \div 2$
Divide $10$ by $2$ to check whether any remainder is obtained.
$\require{enclose}
\begin{array}{rll}
5 && \hbox{} \\[-3pt]
2 \enclose{longdiv}{~10}\kern-.2ex \\[-3pt]
\underline{-~~~10} && \longrightarrow && \hbox{$2 \times 5 = 10$} \\[-3pt]
\phantom{00} 0 && \longrightarrow && \hbox{No Remainder}
\end{array}$
When $10$ is divided by $2$, there is no remainder. This means that $2$ divides $10$ exactly. Therefore, $2$ is a factor of $10$.
Proof
$10 \div 3$
Divide $10$ by $3$ to check whether any remainder is obtained.
$\require{enclose}
\begin{array}{rll}
3 && \hbox{} \\[-3pt]
3 \enclose{longdiv}{~10}\kern-.2ex \\[-3pt]
\underline{-~~~9} && \longrightarrow && \hbox{$3 \times 3 = 9$} \\[-3pt]
\phantom{00} 1 && \longrightarrow && \hbox{Remainder}
\end{array}$
When $10$ is divided by $3$, there is a remainder. This means that $3$ does not divide $10$ exactly. Therefore, $3$ is not a factor of $10$.
Similarly, divide $10$ by $4$ to check whether any remainder is obtained.
Proof
$10 \div 4$ $\,\,\, \longrightarrow \,\,\,$ $\hbox{Remainder 2}$
When $10$ is divided by $4$, there is a remainder. This means that $4$ does not divide $10$ exactly. Therefore, $4$ is not a factor of $10$.
Proof
$10 \div 5$
Divide $10$ by $5$ to check whether any remainder is obtained.
$\require{enclose}
\begin{array}{rll}
2 && \hbox{} \\[-3pt]
5 \enclose{longdiv}{~10}\kern-.2ex \\[-3pt]
\underline{-~~~10} && \longrightarrow && \hbox{$5 \times 2 = 10$} \\[-3pt]
\phantom{00} 0 && \longrightarrow && \hbox{No Remainder}
\end{array}$
When $10$ is divided by $5$, there is no remainder. This means that $5$ divides $10$ exactly. Therefore, $5$ is a factor of $10$.
Similarly, let’s divide $10$ by the numbers from $6$ to $9$ to check whether any remainder is obtained.
Proof
$10 \div 6$ $\,\,\, \longrightarrow \,\,\,$ $\hbox{Remainder 4}$
$10 \div 7$ $\,\,\, \longrightarrow \,\,\,$ $\hbox{Remainder 3}$
$10 \div 8$ $\,\,\, \longrightarrow \,\,\,$ $\hbox{Remainder 2}$
$10 \div 9$ $\,\,\, \longrightarrow \,\,\,$ $\hbox{Remainder 1}$
When $10$ is divided by the numbers from $6$ to $9$ respectively, there is a remainder. This means that the numbers from $6$ to $9$ do not divide $10$ exactly. Therefore, the numbers from $6$ to $9$ are not a factor of $10$.
Proof
$10 \div 10$
Finally, divide $10$ by itself to check whether any remainder is obtained.
$\require{enclose}
\begin{array}{rll}
1 && \hbox{} \\[-3pt]
10 \enclose{longdiv}{~10}\kern-.2ex \\[-3pt]
\underline{-~~~10} && \longrightarrow && \hbox{$10 \times 1 = 10$} \\[-3pt]
\phantom{00} 0 && \longrightarrow && \hbox{No Remainder}
\end{array}$
When $10$ is divided by itself, there is no remainder. This means that $10$ divides itself exactly. Therefore, $10$ is a factor of itself.
What are the Negative Factors of 10?
The positive factors of $10$ are $1$, $2$, $5$, and $10$, and the corresponding negative factors are $-1$, $-2$, $-5$, and $-10$.
What are the Factor Pairs of 10?
The factors of $10$ can be written in pairs so that the product of each pair equals $10$, and they can be arranged into two distinct pairs.
- The positive factor pair of the number $10$ are $(1, 10)$ and $(2, 5)$.
- The negative factor pair of the number $10$ are $(-1, -10)$ and $(-2, -5)$.
How to Factorize 10
The number $10$ can be written in four ways as a product of its factors using factor pairs.
- $10$ $\,=\,$ $1 \times 10$
- $10$ $\,=\,$ $2 \times 5$
- $10$ $\,=\,$ $(-1) \times (-10)$
- $10$ $\,=\,$ $(-2) \times (-5)$
Representation of Factors of 10
The factors of $10$, including both positive and negative integers, can be written in set notation as follows,
$F_{10} = \{-10, -5, -2, -1, 1, 2, 5, 10\}$
