All whole numbers except for 1 are either prime or composite, and composite numbers can always be written as a product of primes. Finding this product (called a prime factorisation) can be very useful.
A number is prime if it has exactly two factors: 1, and itself. A number is composite if it has more than two factors. Learn more about prime numbers in our investigation:  The Sieve of Eratosthenes .
One of the best ways to find a prime factorisation is by using a factor tree. We start with the number we want to investigate, draw a box around it, and draw two lines coming out of it.
Even though the number in the box is different, the numbers at the end of the branches will always be the same for any number - they will just be in a different order.
We can therefore write:
\displaystyle 360 | \displaystyle = | \displaystyle 3 \times 5 \times 2 \times 2 \times 2 \times 3 | |
\displaystyle 360 | \displaystyle = | \displaystyle 2 \times 2 \times 2 \times 3 \times 3 \times 5 | Rewrite the factors in ascending order |
\displaystyle 360 | \displaystyle = | \displaystyle 2^{3} \times 3^{2} \times 5 | Use index form |
Notice that the factor tree for 12 we made earlier is a smaller part of the factor tree for 360. This is because 12 is a factor of 360, and when we write 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 we can recognize the prime factorisation of 12 inside it: 360 = 2 \times (2 \times 2 \times 3) \times 3 \times 5.
Try the applet below to practise forming factor trees. Use the check box to know if your answer is correct or if you need a hint.
The branch of a factor tree stops where the factor is a prime number. We do not include 1 as part of our factor tree.
A number has the following factor tree:
What is this number at the top of the tree?
Write 148 as a product of prime factors in expanded form.
A number is a prime if it has exactly two factors: 1, and itself.
A number is composite if it has more than two factors.
Factor tree starts with the number that needs to be investigated and branches out to two factors. Each composite number in the factor tree branches out to two more factors until the last row of the tree are all primes.
Factor trees are useful because every number we write as we make it is a factor of the original number. We do not always see every factor appear, though - for example, 9 is a factor of 360, but it does not appear in the tree above.
To find every factor of a number we need to combine the prime factors in every possible way. First, we find the prime factorisation like we did before, such as:
12 = 2 \times 2 \times 3
We then combine these factors in every possible way. Every factor of 12 can have no 2s, one 2, or two 2s in its prime factorisation. Similarly, every factor of 12 can have no 3s, or one 3. Here we draw this out in a table:
\text{No }2 | \text{One }2 | \text{Two }2\text{s} | |
---|---|---|---|
\text{No }3 | 1 | 2 | 2 \times 2 |
\text{One }3 | 3 | 2 \times 3 | 2 \times 2 \times 3 |
We then perform each of the multiplications to find all the factors:
\text{No }2 | \text{One }2 | \text{Two }2\text{s} | |
---|---|---|---|
\text{No }3 | 1 | 2 | 4 |
\text{One }3 | 3 | 6 | 12 |
The factors of 12 are 1,\,2,\,3,\,4,\,6, and 12.
Here is how we can do it for 360 - every factor either has 5 as a factor or it doesn't, it has between zero and two 3s, and between zero and three 2s.
\text{ } | \text{No }2 | \text{One }2 | \text{Two }2\text{s} | \text{Three }2\text{s} | |
---|---|---|---|---|---|
\text{No }5 | \text{No }3 | 1 | 2 | 2 \times 2 | 2 \times 2 \times 2 |
\text{No }5 | \text{One }3 | 3 | 2 \times 3 | 2 \times 2 \times 3 | 2 \times 2 \times 2 \times 3 |
\text{No }5 | \text{Two }3\text{s} | 3 \times 3 | 2 \times 3 \times 3 | 2 \times 2 \times 3 \times 3 | 2 \times 2 \times 2 \times 3 \times 3 |
\text{One }5 | \text{No }3 | 5 | 2 \times 5 | 2 \times 2 \times 5 | 2 \times 2 \times 2 \times 5 |
\text{One }5 | \text{One }3 | 3 \times 5 | 2 \times 3 \times 5 | 2 \times 2 \times 3 \times 5 | 2 \times 2 \times 2 \times 3 \times 5 |
\text{One }5 | \text{Two }3\text{s} | 3 \times 3 \times 5 | 2 \times 3 \times 3 \times 5 | 2 \times 2 \times 3 \times 3 \times 5 | 2 \times 2 \times 2 \times 3 \times 3 \times 5 |
This table shows all the possible ways to multiply the prime factors together. We evaluate the multiplications to find all the factors:
\text{ } | \text{No }2 | \text{One }2 | \text{Two }2\text{s} | \text{Three }2\text{s} | |
---|---|---|---|---|---|
\text{No }5 | \text{No }3 | 1 | 2 | 4 | 8 |
\text{No }5 | \text{One }3 | 3 | 6 | 12 | 24 |
\text{No }5 | \text{Two }3\text{s} | 9 | 18 | 36 | 72 |
\text{One }5 | \text{No }3 | 5 | 10 | 20 | 40 |
\text{One }5 | \text{One }3 | 15 | 30 | 60 | 30 |
\text{One }5 | \text{Two }3\text{s} | 45 | 90 | 180 | 360 |
The factors of 360 are 1, 2,\,3,\,4,\,5,\,6,\,8,\,9,\,10,\,12,\,15,\,18,\,20,\,24,\,30,\,36,\,40,\,45,\,60,\,72,\,90,\,120,\,180 and 360.
Find the factors of 20.
Write 20 as a product of prime factors in expanded form.
Using your answer from part (a), list all the factors of 20.
We can multiply all combinations of the prime factors of a number in order to find all the factors of the same number.