What is Prime Factorization
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Example 1) Complete the factor tree and write down the prime factorization of 12.

• The number in the top rectangle is the number we are trying to find the factors for.
• The numbers in the circles are the prime factors that multiply together to give the number above.
• The numbers in the other rectangles are composite (non-prime) numbers that we still need to find the factors for.

We know that 12 = 2 x 6 and 12 = 3 x 4 (we cannot use 1 in a factor tree!)

We can use either of these equations - it does not matter which one.

Let's use 12 = 2 x 6. So the prime factor is 2 and the composite factor is 6.

This gives us:

Now we only have the 6 left to factorize.

We know that 6 = 1 x 6 and 6 = 2 x 3.

We cannot use 1 in a factor tree, so that leaves us with 6 = 2 x 3.

This gives us:

The prime factors of 12 are 2, 2, and 3.

This means that 12 = 2 x 2 x 3 (or 2² x 3) as a product of prime factors (prime factorization).

Example 2) Let us go back to Example 1) and factorize it in a different way.

This time we will factorize 12 as 3 x 4 instead of 2 x 6.

This gives us:

Now we only have the 4 left to factorize.

We know that 4 = 1 x 4 and 4 = 2 x 2.

We cannot use 1 in a factor tree, so that leaves us with 4 = 2 x 2.

This gives us:

The prime factors of 12 are still 2, 2, and 3.

So we still end up with 12 = 2 x 2 x 3 (or 2² x 3) as a product of prime factors (prime factorization.

So we still have the same answer as Example 1) but the factor trees have some different numbers in.

The Factor Tree in Example 1) has a 6, and the Factor Tree in Example 2) has a 4.

This does not matter - the important thing is that the numbers in the circles (the prime factors) are the same in both examples.

Example 3) Draw a factor tree to find the prime factorization of 90.

We know that 2 is a factor of 90 as it is even.

90 = 2 × 45

2 is prime, but 45 is not. This gives us:

We now need to factorize 45.

We know that 45 = 1 x 45 and also 45 = 3 x 15.

The factorization we need is 45 = 3 x 15.

3 is prime, but 15 is composite.

This gives us:

We now need to factorize 15.

15 = 3 x 5, which are both prime, so we have finished.

Our final factor tree is:

This gives us a final factorization of 90 = 2 x 3 x 3 x 5 or 2 x 3² x 5

Example 1) Find the prime factorization of 40.

40 is even, so we know that 2 is a factor (as well as being prime)

40 = 2 x 20.

Now we move on to the number 20.

Again 20 is even, so 2 is a factor.

20 = 2 x 10

So 40 = 2 x 20 = 2 x 2 x 10

Now we factorize 10 into 2 x 5 (both prime numbers) and we cannot go any further, as all the numbers are prime.

So 40 = 2 x 20 = 2 x 2 x 10 = 2 x 2 x 2 x 5

Our final answer is 40 = 2 x 2 x 2 x 5 = 2³ x 5

Example 2) Find the prime factorization of 45.

45 is odd, so 2 is not a factor.

As the number ends in a 5 we know that 5 is a factor.

45 = 5 x 9.

Now we have to factorize 9.

9 = 3 x 3 (which is a prime number).

This gives us:

45 = 5 x 9 = 5 x 3 x 3

Our final answer is 45 = 3 x 3 x 5 = 3² x 5

Example 3) Find the prime factorization of 126.

126 is even, so 2 is a factor.

126 = 2 x 63

Now we need to factorize 63.

63 = 7 x 9.

7 is prime, but 9 is not

Now we have to factorize 9.

9 = 3 x 3 (which is a prime number).

This gives us:

126 = 2 x 63 = 2 x 7 x 9 = 2 x 7 x 3 x 3

Our final answer is 126 = 2 x 3 x 3 x 7 = 2 x 3² x 7