Multiples, factors and prime factors

​​In a nutshell

Every number is either a prime number, or a multiple of other factors.

Multiples, factors and primes

• Multiple - A multiple of a number is any number that can be formed as a product of that number and another number.
  
• Factor - A factor of a number is any number that divides the number with no remainder.
• Prime number - A prime number is a number which has only two distinct factors, $$1$$ and itself.​

Example 1

What are the factors of $$36$$?

$$36 = 1 \times 36$$

$$36 = 2 \times 18$$​​​

$$36 = 3 \times 12$$

$$36 = 4 \times 9$$​​​

$$36 = 6\times 6$$

Hence the factors of $$36$$ are $$\underline{1,2,3,4,6,9,12,18,36}$$.

Example 2

Is $$162$$ a multiple of $$27$$?

$$162 \div 27 = 6$$

Yes, $$162$$ is a multiple of $$27$$, as the quotient ($$6$$) is a whole number and there is no remainder.

Factor trees

You can find the prime factorisation of a number using a factor tree.

Procedure

1. ​Write the number at the top and find two numbers which multiply to make the number.
   
2. Write the pair of factors underneath the original number, "branching" them off to the left and right of the original number.
   
3. If one, or both, of these factors are not prime, repeat this process until the end of every "branch" is a prime number.
   
4. Write the numbers at the bottom of each branch, multiplied by one another. This is the prime factorisation.
   

Example 3

Use a factor tree to find the prime factorisation of $$273$$.

​$$\quad\quad\quad\space{\begin{aligned} &\space \space \,\,273\\\ &\,\,\swarrow\searrow& \\&\textcircled{3}\,\,\,\,\, \quad91 \\&\quad\,\,\swarrow\searrow\\&\space\space\space\,\,\textcircled7 \,\,\,\,\quad\textcircled{13} \\\end{aligned}}$$​​

Hence the prime factorisation of $$273$$ is $$\underline{273 = 3 \times 7 \times 13}$$.

Note: If the original number is, itself, a prime number then you already have the number in its prime factorisation form.