Multiples, factors and prime factors

​​In a nutshell

Multiples and factors are both very useful applications of multiplication and division. Furthermore, numbers can be broken down into their prime factors, which has useful applications.

Multiples and factors

Definitions

A multiple of a number is an integer multiplied by the number.

A factor of a number divides into the number without any remainders.

Examples

• ​$$20$$ is a multiple of $$5$$ because $$20=4\times5.$$
• $$4$$ is a factor of $$100$$ because $$100\div4=25$$.
• $$18$$ is not a multiple of $$10$$ because $$18$$ is not in the $$10$$ times tables.
• ​$$8$$​ is not a factor of $$12$$ because $$12\div8=1$$ remainder $$4$$.

​​Finding factors of a number

To find all the factors of a number, follow this procedure.

Procedure

1. ​Write the number as $$1\times$$ itself.
2. Move onto the next number ($$2$$), and find the pair that multiplies by $$2$$ by writing the number as $$2\times \_$$ (if applicable).
3. Move onto the next number, and repeat the process until you get repeated pairs.
   
4. The factors are all the numbers that have a pair where both numbers are integers.
   

Example 1

Find all the factors of $$18$$.

Start from $$1$$:

​$$18=1\times18$$​

Hence, $$1$$ and $$18$$​ are factors.

Continue in the same fashion with $$2,3,4...$$ until you get a repeated pair.

$$18=2\times9$$​​

$$18=3\times6$$​​

$$18=4\times4.5$$​​

$$18=5\times3.6$$​​

$$18=6\times3$$​​

This is a repeated pair ($$3\times6$$ and $$6\times3$$). So, stop here.

The factors of $$18$$ are $$\underline{1,2,3,6,9,18}$$.

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Prime factors

A prime factor of a number is a factor that is also prime. ​​

Prime factor decomposition

Every positive integer can be broken down into a unique product of prime factors. This means to write a number as a prime number or numbers being multiplied together. For example, the number $$18$$​ can be written as $$3\times3\times2$$​, or $$3^2\times2$$​. This is called the prime factorisation of a number. To find a number's prime factorisation, you can use a factor tree.

procedure

1. ​Write down the number at the top. Find two numbers that multiply to give the number.
   
2. Write down these two factors, branching off to the bottom left and bottom right of the original number.
   
3. Proceed similarly with each individual factor, until each "branch" results in a prime factor.
   
4. Circle each prime factor, and write them all multiplied by each other. This is the prime factorisation.
   

Example 2

Find the prime factorisation of $$180$$.

Write $$180$$​ on the top, and find two numbers that multiply to give $$180$$:

​$$180=18\times10$$​​

Write $$18$$ and $$10$$ beneath $$180$$, and do the same for each factor.

For $$10$$:

​$$10=\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {5}}}\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {2}}}$$​​

Note that $$5$$ and $$2$$ are both prime numbers, so circle them and move on to $$18$$.

For $$18$$:

​$$18=9\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {2}}}$$​​

Note that $$2$$ is prime, so circle it and move on to $$9$$.

For $$9$$:

​$$9=\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {3}}}\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {3}}}$$​​

​$$3$$​ is prime, so circle both the threes.

Now, write down all the circled numbers (these are the prime factors) multiplied by each other:

$$180=2\times2\times3\times3\times5$$​​

Write the repeated factors as indices:

$$\underline{180=2^2\times3^2\times5}$$

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