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Multiples, factors and prime factors

Multiples, factors and prime factors

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Summary

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
  • 2020 is a multiple of 55 because 20=4×5.20=4\times5.
  • 44 is a factor of 100100 because 100÷4=25100\div4=25.
  • 1818 is not a multiple of 1010 because 1818 is not in the 1010 times tables.
  • 88is not a factor of 1212 because 12÷8=112\div8=1 remainder 44.


​​Finding factors of a number

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


Procedure

  1. ​Write the number as 1×1\times itself.
  2. Move onto the next number (22), and find the pair that multiplies by 22 by writing the number as 2×_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 1818.


Start from 11:

18=1×1818=1\times18


Hence, 11 and 1818 are factors.


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

18=2×918=2\times9​​

18=3×618=3\times6​​

18=4×4.518=4\times4.5​​

18=5×3.618=5\times3.6​​

18=6×318=6\times3​​


This is a repeated pair (3×63\times6 and 6×36\times3). So, stop here.


The factors of 1818 are 1,2,3,6,9,18\underline{1,2,3,6,9,18}.


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 1818​ can be written as 3×3×23\times3\times2​, or 32×23^2\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 180180.


Write 180180​ on the top, and find two numbers that multiply to give 180180:

180=18×10180=18\times10​​


Write 1818 and 1010 beneath 180180, and do the same for each factor.

For 1010:

10=5×210=\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {5}}}\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {2}}}​​


Note that 55 and 22 are both prime numbers, so circle them and move on to 1818.


For 1818:

18=9×218=9\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {2}}}​​


Note that 22 is prime, so circle it and move on to 99.


For 99:

9=3×39=\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {3}}}\times\raisebox{.5pt}{\textcircled{\raisebox{-.6pt} {3}}}​​


33is prime, so circle both the threes.


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

180=2×2×3×3×5180=2\times2\times3\times3\times5​​


Write the repeated factors as indices:

180=22×32×5\underline{180=2^2\times3^2\times5}

​​

Maths; Types of numbers; KS3 Year 7; Multiples, factors and prime factors

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