Multiples
A multiple of a number is a number which occurs in that number's times table.
eg. multiples of 3 are 3, 6, 9, 12, 15...
Note that all numbers have an infinite number of multiples which are all bigger than or equal to the number itself.
Also note that 3 is the first multiple of 3 as it is the answer to the multiplication 3 × 1.
Factors
Factors are numbers which can be divided exactly, into a number. This means that factors are always equal to or smaller than the number itself.
The factors of 6 are all of the numbers which will divide into 6 exactly, without leaving a remainder.
So the factors of 6 are 1, 2, 3, 6.
Please have a look at the Worked examples tab, above, for more details on finding factors and multiples and related topics.
Worked example 1: Multiples of 4
- Notes
What are the first 5 multiples of 4?When we talk of the first 5 multiples of a number, we will always begin with $1 \times$ the number.
\[\begin{aligned} 1 \times 4 &=4\\ 2 \times 4 &=8\\ 3 \times 4 &=12\\ 4 \times 4 &=16\\ 5 \times 4 &=20\\ \end{aligned}\]You need to watch out here as it is very easy to mistake the number of items in the list with the number you are finding multiples of. In this case the numbers 4 and 5. This is a very common careless error.You will know when you have the completed list when you reach the 5th multiple of 4; ie $5 \times 4$.
The first 5 multiples of 4 are 4, 8, 12, 16, 20.
Worked example 2: All the factors of 15
- Notes
- Write down all of the factors of 15.
Start by writing down $1 \times $ the number you wish to factorise, in this case 15. Then run through numbers which may divide into 15. 2 clearly doesn't as 15 is an odd number, but 3 does, 5 times. So we can write down $3 \times 5$. 4 doesn't, for the same reason as 2, but 5 does. But here is the trick, we have already written down 5, so our list is complete. This will always work to give you the full list.
$$\begin{aligned} &1 \times 15 \\ &3 \times 5\\ \end{aligned}$$You should then read off the list using a U shape from left to right, start at 1, 3 and then going up to 5 and 15.
Finding factors is a technique which all school children learn and seasoned mathematicians are still grappling with. The only difference is the size of the numbers involved. Finding the factors of 15 efficiently, is pretty straightforward, but if you are trying to find the factors of a numbers with many thousands of digits, not so much! However, let us deal with this simple example.- The factors of 15 are 1, 3, 5, 15
Worked example 3: All the factors of 144
- Notes
- Write down all of the factors of 144.
- $$\begin{aligned} &1 \times 144 \\ &2 \times 72\\ &3 \times 24\\ &4 \times 36\\ &6 \times 24\\ &8 \times 18\\ &9 \times 16\\ &12 \times 12 \end{aligned}$$
Start by writing down $1 \times 144$. Then work your way through the numbers. Ask yourself why you only need to go up to $12$. $16$ is the next factor of $144$, after $12$. You should notice that $16$ is already in our list.
Effectively what this means is that you only need to try numbers up to the square root of the original number. After that, you will always get repeats. - The factors of 144 are:
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 72, 144
Exercise 1: Finding multiples
- Q1Write down the first 5 multiples of 6.
6, 12, 18, 24, 30 - Q2Write down the first 3 multiples of 4.
4, 8, 12
- Q3Write down the first 4 multiples of 9.
9, 18, 27, 36
- Q4Which of the following numbers are multiples of 3?
- 45
- 56
- 72
- 1326
45, 72, 1326
are multiples of 3.
(56 is not.) - Q5Find the first 2 common multiples of 3 and 5.
15, 30
- Q6Find the first three common multiples of 6 and 9.
18, 36, 54
- Q7
Copy and complete the following sentences, using one of the words in the cloud. Please note that there is only one correct answer to each.
- 5 is a ................of 45.
- 36 is a ................of 9.
- 54 is the ................of 9 and 6.
- 2 is the first...............
- Factor
- Multiple
- Product
- Prime
- Q8a. Find the first two common multiples of 12 and 9.
b. Which of these is the Lowest Common Multiple?
a. 36, 72.
b. 36 - Q9a. Write out the first 10 multiples of 6 and 4.
b. Ring the common multiples of 6 and 4.
c. Which is the Lowest Common Multiple of 6 and 4.
a. 4, 8, 12, 16, 20, 24, 28, 32, 36
b. 6, 12, 18, 24, 30, 36, 42, 48, 54.
c. LCM(4, 6) = 12
Exercise 2: Finding factors
- Q1Find all the factors pairs of 6.
$\begin{aligned}{1 \times 6}& \\{2 \times 3}&\end{aligned}$
- Q2Find all the factor pairs of 12
$\begin{aligned} {1 \times 12}& \\{2 \times 6}& \\{3 \times 4} \end{aligned}$
- Q3Find all the factors of 42.
$\begin{aligned} {1 \times 42}& \\{2 \times 21}& \\{3 \times 14}& \\{6 \times 7} \end{aligned}$
- Q4Find all the factors of 54.
$\begin{aligned} {1 \times 54}& \\{2 \times 27}& \\{3 \times 18}& \\{6 \times 9} \end{aligned}$
- Q5Find all the factors of 72.
$\begin{aligned} {1 \times 72}& \\{2 \times 36}& \\{3 \times 24}& \\{4 \times 18}& \\{6 \times 12}& \\{8 \times 9} \end{aligned}$
- Q6Find all the factors of 56.
$\begin{aligned} {1 \times 56}& \\{2 \times 28}& \\{4 \times 14}& \\{7 \times 8}& \end{aligned}$
- Q7Find all the factors of 32.
$\begin{aligned} {1 \times 32}& \\{2 \times 16}& \\{4 \times 8} \end{aligned}$
- Q8Find all the factors of 96.
$\begin{aligned} {1 \times 96}& \\{2 \times 48}& \\{3 \times 32}& \\{4 \times 24}& \\{6 \times 16}& \\{8 \times 12} \end{aligned}$
- Q9Find all the factors of 1024.
$\begin{aligned} {1 \times 1024}& \\{2 \times 512}& \\{4 \times 256}& \\{8 \times 128}& \\{16 \times 64}& \\{32 \times 32} \end{aligned}$
- Q10a. Find all the factor pairs of the following numbers
- 4
- 9
- 16
- 25
c. Do you notice anything about the number of factors each of these numbers have?
d. Can you explain why this is?
-
- Factors of 4 are
$\begin{aligned}{1 \times 4}& \\{2 \times 2} \end{aligned}$ - Factors of 9 are
$\begin{aligned}{1 \times 9}& \\{3 \times 3} \end{aligned}$ - Factors of 16 are
$\begin{aligned}{1 \times 16}& \\{2 \times 8}& \\{4 \times 4}\end{aligned}$ - iv. Factors of 25 are
$\begin{aligned}{1 \times 25}& \\{5 \times 5} \end{aligned}$
- Factors of 4 are
- They are all square numbers.
- All have an odd number of factors.
- This is because square numbers all have a repeated factor, which will only be counted once.