Extension: Prime factorizations

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 factoring) can be very useful.

Factor trees

One of the best ways to find a prime factoring 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. Here is how we might start with the number $$12:

We then put two numbers that multiply to make $$12, such as $$4 and $$3, at the end of each of the lines:

Because $$3 is a prime number, we circle it. Since $$4 is not a prime number, we draw a box around it instead:

We then repeat the process with $$4, which is $$2×2:

And since $$2 is prime, we circle both of these numbers:

This is a completed factor tree for $$12, and it tells us that $$12=2×2×3. Multiplying the circled numbers at the end of each branch together always makes the original number.

Factor trees are not always unique - here is another factor tree for $$12:

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.

Here is a factor tree for $$360:

We can therefore write: 

$$360=3×5×2×2×2×3

We usually rewrite the factors so they are in ascending order, like this:

$$360=2×2×2×3×3×5

We can use exponent notation to make the expression shorter as well:

$$360=23×32×5

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 $$360=2×2×2×3×3×5 we can recognize the prime factoring of $$12 inside it: $$360=2×(2×2×3)×3×5.

 

Finding all the factors

Factor trees are useful because every number we write as we make it is a factor of the original number. We don't 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 factoring like we did before, such as:

$$12=2×2×3

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 factoring. Similarly, every factor of $$12 can have no $$3s, or one $$3. Here we draw this out in a table:

Factors of $$12
	No $$2	One $$2	Two $$2s
No $$3	$$1	$$2	$$2×2
One $$3	$$3	$$2×3	$$2×2×3

We then perform each of the multiplications to find all the factors:

Factors of $$12
	No $$2	One $$2	Two $$2s
No $$3	$$1	$$2	$$4
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.

Factors of $$360
		No $$2	One $$2	Two $$2s	Three $$2s
No $$5	No $$3	$$1	$$2	$$2×2	$$2×2×2
	One $$3	$$3	$$2×3	$$2×2×3	$$2×2×2×3
	Two $$3s	$$3×3	$$2×3×3	$$2×2×3×3	$$2×2×2×3×3
One $$5	No $$3	$$5	$$2×5	$$2×2×5	$$2×2×2×5
	One $$3	$$3×5	$$2×3×5	$$2×2×3×5	$$2×2×2×3×5
	Two $$3s	$$3×3×5	$$2×3×3×5	$$2×2×3×3×5	$$2×2×2×3×3×5

This table shows all the possible ways to multiply the prime factors together. We evaluate the multiplications to find all the factors:

Factors of $$360
		No $$2	One $$2	Two $$2s	Three $$2s
No $$5	No $$3	$$1	$$2	$$4	$$8
	One $$3	$$3	$$6	$$12	$$24
	Two $$3s	$$9	$$18	$$36	$$72
One $$5	No $$3	$$5	$$10	$$20	$$40
	One $$3	$$15	$$30	$$60	$$120
	Two $$3s	$$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.

 

Practice questions

Question 1

Question 2

Question 3