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