What are the prime factors of 12?

It is best to start working from the smallest prime number, which is 2, so let’s check:

12 ÷ 2 = 6

Yes, it is divided exactly by 2. We have taken the first step!

But 6 is not a prime number, so we need to go further. Let’s try 2 again:

6 ÷ 2 = 3

Yes, that worked also. And 3 is a prime number, so we have the answer:

12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right.

Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 = 2² × 3

What is the prime factorization of 147?

Can we divide 147 exactly by 2?

147 ÷ 2 = 73½

No, it can’t. The answer should be a whole number, and 73½ is not.

Let’s try the next prime number, 3:

147 ÷ 3 = 49

That worked, now we try factoring 49.

The next prime, 5, does not work. But 7 does, so we get:

49 ÷ 7 = 7

And that is as far as we need to go because all the factors are prime numbers.

147 = 3 × 7 × 7 (or 147 = 3 × 7² using exponents)

Example:

What are the prime factors of 90?

Break 90 into 9 × 10

• The prime factors of 9 are 3 and 3
• The prime factors of 10 are 2 and 5

So the prime factors of 90 are 3, 3, 2 and 5

Prime Factorization by Tree Method

Factor trees are a way of expressing the factors of a number, specifically the prime factorization of a number. 

Each branch in the tree is split into factors. Once the factor at the end of the branch is a prime number, the only two factors are itself and one so the branch stops and we circle the number.

We also must remember that 1 is not a prime number and so it will not appear in any factor tree.

Factor trees can be used to:

• find the highest common factor (HCF),
• find the lowest common multiple (LCM) (sometimes called the least common multiple)
• find other numerical properties such as whether a number is square, cube, or prime

Prime Factor Tree

To produce a prime factor tree, we need to be able to recall the prime numbers between 1 and 20.

Let’s have a look at an example:
Example: Use a factor tree to write 51 as a product prime factors

We split the original number 51 into two branches by writing a pair of factors at the end of the branch,
As 3 × 17 = 51, one branch will end in a 3, the other in 17.

Both the numbers 3 and 17 are prime numbers and so we highlight the prime numbers by circling them.

Now there is a prime number at the end of each branch we have constructed a prime factor tree.
If the numbers were not primes then we would continue to split them into factors until there was a prime number at the end of each branch.

We can now write 51 as a product of its prime factors by using the numbers at the ends of the branches: $$51=3\times 17$$

Tip: Write the prime factors in order, smallest to largest.

Remember:

The factor trees of a number are not unique, but the product of prime factors is unique.
This means that a number could have multiple different factor trees that will all give the same product of prime factors.

Writing An answer In Index Form

Writing a number as a product of its prime factors we should write it in index form.

Example: Express the number 24 as a product of prime factors

So,

We can write this in index form:

By using an alternative pair of factors for 24, we can see that even though the factor tree is different, the same unique prime factorization of 24 is given.

So,

We can write this in index form:

How to use a factor tree:

To use a factor tree:

1. Write the number at the top of the factor tree and draw two branches below
2. Fill in the branches with a factor pair of the number above
3. Continue until each branch ends in a prime number
4. Write the solution as a separate line of working (in index form if required)

Common Misconceptions

• Using addition instead of multiplication

When creating a factor tree for say 26, a common mistake is to write the factors of 26 as 13 and 13. This is incorrect as 13 × 13 = 169 giving the prime factor decomposition of 169, not 26.

• Assuming a number is prime

There are several numbers that are frequently misused as a prime number, here are a few of them:
1, 9, 15, 21, 27They are usually a multiple of 3 unless they are more difficult to split into factors, such as 57 and 91. (57 = 3 × 19, 91 = 7 × 13).

• Not writing the final solution

After completing the factor tree, you must write the number as a product of its factors, otherwise, you have demonstrated a method but not answered the question (such as using grid multiplication and not adding up the values in the grid for your final solution).

• Incorrect simplifying of solution

Once you have reached a prime number in the factor tree, highlight it, otherwise, it can get lost in the complexity of the factor tree.
Space out the diagram so you can clearly see all the factors and circle the prime factors for your solution. Then carefully check how many of each prime number exist, then write the solution using index form. The order of the product of prime factors does not matter but the numbers do!