Factors and Multiples
We learn how to find the multiples of whole numbers as well as how to find a number's factors.
Definition - Multiples
Given an integer, for example \(3\), the multiples of \(3\) are all the integers we obtain if we multiply \(3\) by the positive integers: \(1\), \(2\), \(3\), ... .
So the multiples of \(3\) are:
\[\begin{aligned} & 1 \times 3 = 3\\
& 2 \times 3 = 6 \\
& 3 \times 3 = 9 \\
& 4 \times 3 = 12\\
& \dots
\end{aligned} \]
A more general definition of the multiples of an arbitrary integer \(n\) would be:
\[k \times n\]
Where \(k\) is a positive integer.
Tutorial 1: Multiples
In the following tutorial we review the ... .
Exercise 1
Write down the first \(5\) multiples of \(3\).
What is the \(7^{\text{th}}\) multiple of \(2\)?
State the first \(4\) multiples of \(7\).
What is the \(6^{\text{th}}\) multiple of \(4\)?
Write down the first \(6\) multiples of \(5\).
Answers Without Working
The first \(5\) multiples of \(3\) are:
\(3, 6, 9, 12, 15\)
The seventh multiple of \(2\) is \(14\).
The first \(4\) multiples of \(7\) are: \(7,14,21,28\).
The \(6^{\text{th}}\) multiple of \(4\) is \(24\).
The first \(6\) multiples of \(5\) are: \(5,10,15,20,25,30\).
Solution With Working
The first \(5\) multiples of \(3\) are:
\(1\times 3 = 3\)
\(2\times 3 = 6\)
\(3\times 3 = 9\)
\(4\times 3 = 12\)
\(5\times 3 = 15\)
The \(7^{\text{th}}\) multiple of \(2\) is:
\(7\times 2 = 14 \)
The first \(4\) multiples of \(7\) are:
\(1\times 7 = 7\)
\(2\times 7 = 14\)
\(3\times 7 = 21\)
\(4\times 7 = 28\)
The \(6^{\text{th}}\) multiple of \(4\) is: \(6 \times 4 = 24\).
The first \(6\) multiples of \(5\) are:
\(1\times 5 = 5\)
\(2\times 5 = 10\)
\(3\times 5 = 15\)
\(4\times 5 = 20\)
\(5\times 5 = 25\)
\(6\times 5 = 30\)
Exercise 2
State whether each of the following statements is true or false.
Statement: \(12\) is a multiple of \(3\).
Statement: \(7\) is a multiple of \(1\).
Statement: The \(4^{\text{th}}\) multiple of \(5\) is \(20\).
Statement: The \(3^{\text{rd}}\) multiple of \(7\) is \(23\).
Statement: The \(3^{\text{rd}}\) multiple of \(4\) equals to the \(4^{\text{th}}\) multiple of \(3\).
Solution Without Working
true
true
true
false
true
Solution With Working
True, since \(12 = 4\times 3\). \(12\) is the \(4^{\text{th}}\) multiple of \(3\).
True, since \(7 = 7 \times 1\). \(7\) is the \(7^{\text{th}}\) multiple of \(1\).
True, since \(4\times 5 = 20\), the \(4^{\text{th}}\) multiple of \(5\) is \(20\).
False, the \(3^{\text{rd}}\) multiple of \(7\) is \(3\times 7 = 21\), which isn't equal to \(23\).
True, the \(3^{\text{rd}}\) multiple of \(4\) is \(3\times 4 = 12\) and the \(4^{\text{th}}\) multiple of \(3\) is \(4\times 3 = 12\).
Definition - Factors
Given an integer \(p\), we say \(n\) is a factor of \(p\) if and only if \(p\) is a multiple of \(n\).
For instance, \(3\) is a factor of \(6\) because \(6\) is a multiple of \(3\).
Indeed we can write:
\[6 = 2\times 3\]
Note: That also shows us that \(2\) is a factor of \(6\) because \(6\) is a multiple of \(2\), which highlights an important fact: we find factors in pairs. In other words, we always find factors two at a time.
Tutorial 2: Factors
In the following tutorial we learn what a factor is as well as see how to find a number's factors.
Exercise 3
Using the method we just learnt, find the following factors:
factors of \(8\)
factors of \(9\)
factors of \(10\)
factors of \(12\)
factors of \(14\)
factors of \(20\)
Solution Without Working
The factors of \(8\) are:
\[1,2,4,8\]
The factors of \(9\) are:
\[1,3,9\]
The factors of \(10\) are:
\[1,2,5,10\]
The factors of \(12\) are:
\[1,2,3,4,6,12\]
The factors of \(14\) are:
\[1,2,7,14\]
The factors of \(20\) are:
\[1,2,4,5,10,20\]
Solution With Working
To find the factors of \(8\) we work our way through the whole numbers less than or equal to \(8\div 2 = 4\) and check if \(8\) is a multiple:
\(1\): \(8\) is a multiple of \(1\), indeed: \(8 = 8\times 1\), so \(1\) is a factor of \(8\) and so is \(8\).
\(2\): \(8\) is a multiple of \(2\), indeed: \(8 = 4\times 2\), so \(2\) is a factor of \(8\) and so is \(4\).
\(3\): \(8\) is not a multiple of \(3\), so \(3\) is not a factor of \(8\).
\(4\): just as we saw, when considering the factor \(2\), \(4\) is a factor of \(8\).
So the factors of \(8\) are: \(1,2,4,8\).
To find the factors of \(9\), we work our way through the whole numbers less than or equal to \(9 \div 2 = 4.5\), those are the whole numbers from \(1\) to \(4\), and check if \(9\) is a multiple:
\(1\): \(9\) is a multiple of \(1\), indeed: \(9 = 9 \times 1\), so \(1\) is a factor of \(9\) and so is \(9\).
\(2\): \(9\) isn't a multiple of \(2\), so \(2\) isn't a factor of \(9\).
\(3\): \(9\) is a multiple of \(3\), indeed: \(9 = 3\times 3\), so \(3\) is a factor of \(9\).
\(4\): \(9\) isn't a mutiple of \(4\) so \(4\) isn't a factor of \(9\).
So the factors of \(9\) are \(1\), \(3\) and \(9\).
To find the factors of \(10\), we work our way through the whole numbers less than or equal to \(10 \div 2 = 5\), those are the whole numbers from \(1\) to \(5\), and check if \(10\) is a multiple:
\(1\): \(10\) is a multiple of \(1\), indeed \(10 = 10\times 1\), so \(1\) is a factor and so is \(10\).
\(2\): \(10\) is a multiple of \(2\à, indeed \(10 = 5 \times 2\), so \(2\) is a factor of \(10\) and so is \(5\).
\(3\): \(10\) isn't a multiple of \(3\) so \(3\) isn't a factor of \(10\).
\(4\): \(10\) isn't a multiple of \(4\) so \(4\) isn't a factor of \(10\).
\(5\): jsut as we saw, when studying the factor \(2\), \(5\) is a factor of \(10\).
So the factors of \(10\) are \(1,2,5\) and \(10\).
To find the factors of \(12\), we work our way through the whole numbers less than or equal to \(12 \div 2 = 6\), those are the whole numbers from \(1\) to \(6\), and check if \(12 \) is a multiple:
\(1\): \(12\) is a multiple of \(1\), indeed \(12 = 12 \times 1\), so \(1\) is a factor of \(12\) and so is \(12\).
\(2\): \(12\) is a multiple of \(2\), indeed \(12 = 6 \times 2 \), so \(2\) is a factor of \(12\) and so is \(6\).
\(3\): \(12\) is a multiple of \(3\), indeed \(12 = 4 \times 3 \), so \(3\) is a factor of \(12\) and so is \(4\).
\(4\): just as we saw, when studying the factor \(3\), \(4\) is a factor of \(12\)
\(5\): \(12\) is not a multiple of \(5\), so \(5\) is not a factor or \(12\).
\(6\): just as we saw, when studying the factor \(2\), \(6\) is a factor of \(12\)
So the factors of \(12\) are \(1,2,3,4,6\) and \(12\)
To find the factors of \(14\), we work our way through the whole numbers less than or equal to \(14 \div 2 = 7\), those are the whole numbers from \(1\) to \(7\), and check if \(14\) is a multiple:
\(1\): \(14\) is a multiple of \(1\), indeed \(14 = 14\times 1\), so \(1\) is a factor of \(14\) and so is \(14\).
\(2\): \(14\) is a multiple of \(2\), indeed \(14 = 7 \times 2\), so \(2\) is a factor of \(14\) and so is \(7\)
\(3\): \(14\) is not a multiple of \(3\) so \(3\) is not a factor of \(14\).
\(4\): \(14\) is not a multiple of \(4\) so \(4\) is not a factor of \(14\).
\(5\): \(14\) is not a multiple of \(5\) so \(5\) is not a factor of \(14\).
\(6\): \(14\) is not a multiple of \(6\) so \(6\) is not a factor of \(14\).
\(7\): just as we saw, when studying the factor \(2\), \(7\) is a factor of \(14\).
So the factors of \(14\) are \(1,2,7\) and \(14\).
To find the factors of \(20\), we work our way through the whole numbers less than or equal to \(20 \div 2 = 10\), those are the whole numbers from \(1\) to \(10\), and check if \(20\) is a multiple:
\(1\): \(20\) is a multiple of \(1\), indeed \(20 = 20\times 1\), so \(1\) is a factor of \(20\) and so is \(20\).
\(2\): \(20\) is a multiple of \(2\), indeed \(20 = 10\times 2\), so \(2\) is a factor of \(20\) and so is \(10\).
\(3\): \(20\) is not a multiple of \(3\) so \(3\) is not a factor of \(20\).
\(4\): \(20\) is a multiple of \(4\), indeed \(20 = 5 \times 4\), so \(4\) is a factor of \(20\) and so is \(5\).
\(5\): as we have just seen, when studying the factor \(4\), \(5\) is a factor of \(20\).
\(6\): \(20\) is not a multiple of \(6\) so \(6\) is not a factor of \(20\).
\(7\): \(20\) is not a multiple of \(7\) so \(7\) is not a factor of \(20\).
\(8\): \(20\) is not a multiple of \(8\) so \(8\) is not a factor of \(20\).
\(9\): \(20\) is not a multiple of \(9\) so \(9\) is not a factor of \(20\).
\(10\): as we saw, when considering the factor \(2\), \(10\) is a factor of \(20\).
So the factors of \(20\) are \(1,2,4,5,10\) and \(20\).
Exercise 4
State whether each of the following statements is "true" or "false":
\(9\) is a factor or \(12\).
\(12\) is a factor of \(3\).
\(3\) is a factor of \(12\).
\(8\) is a factor of \(24\).
\(1\) is a factor of all whole numers.
If \(n\) is a factor of \(m\) then \(m\) is a multiple of \(n\).
Solution Without Working
false
false
true
true
true
true
Solution With Working
false: \(9\) is not a factor of \(12\) since \(12\) is not a multiple of \(9\).
false: \(12\) is not a factor of \(3\), it is a multiple of \(3\).
true: \(3\) is a factor of \(12\) since \(12\) is a multiple of \(3\).
true: \(8\) is a factor of \(24\) since \(24\) is a multiple of \(8\).
true: \(1\) is a factor of all whole numbers since all whole numbers are multiples of \(1\).
true: by definition a whole number \(n\) is a factor of another whole number \(m\) if and only if \(m\) is a multiple of \(n\). So if \(n\) is a factor of \(m\) then \(m\) must be a multiple of \(n\).