Prime Factorization: Mastering Division by Primes

Unlock the power of prime factorization with our step-by-step guide. Learn essential techniques, from factor trees to the ladder method, and boost your math prowess. Perfect for students of all levels!

Factor

One of the first things you'll need to learn about before tackling prime and composite numbers are factors. Factors are what you'll have to multiply together in order to get a certain number. So for example, the factors of 18 is 2 and 9 since 2 x 9 = 18. The divisor definition is a number that another number is to be divided with. In this case, 2 and 9 are each also known as a divisor.

Prime numbers

So what are prime numbers? Prime numbers are numbers that can only be factored by 1 and itself. An example of this is 2. You cannot get an answer of 2 in multiplication other than multiplying 2 with 1. 2 is known as a prime divisor.

Composite Numbers

Contrary to prime numbers, composite numbers are numbers that have more factors than just 1 and itself. It also is positive. For example, 9 can be factored into 3 x 3 or it can be 9 x 1. All whole numbers are either prime or composite, other than the number 1 and 0. 0 has an infinite amount of factors, whereas 1 cannot be made up of anything that is not itself.

Factor Tree

To tackle the questions in this lesson, you're going to have to learn about a factor tree. A factor tree breaks down a number so that you're able to identify its prime factors. The steps to making a factor tree is:

1) Write down the number you are trying to factorize at the top of the tree

2) Draw two branches stemming from the number downwards

3) Break down the original number into two factors and write it at the end of the branches you just drew in the previous step

4) Continue breaking down the numbers into factors at the end of branches until you're left with all prime numbers and there are no more factors to be found

5) Take all the numbers at the end of the factor tree branches to find out the prime factors of your original number

Continuous division

You can use a method called continuous division to find the greatest common factor (GCF) of a number. The GCF also deals with prime numbers. You can carry out continuous division by:

1) Writing down the two numbers you're trying to find the GCF of

2) Draw an "L" shape surrounding them

3) Divide both the numbers by a common factor

4) Write the answers you get underneath the bar

5) Continue doing this until all the numbers you're left with as answers are prime numbers

6) Multiply together all the numbers on the left hand side (that has been common factors of the numbers inside the "L") and you'll get your GCF

Example problems

Question 1:

1a) Factors of $12$

Solution:

$12$

$4 \bullet 3$

Since $4$ is not a prime number, we can break it down into two times two

$2 \bullet 2 \bullet 3$

We are done now because all the numbers are prime, but we can group up the same numbers in exponent form.

$2^{2} \bullet 3$

1b) Factors of $24$

Solution:

$24$

$2 \bullet 12$

$2 \bullet 2 \bullet 6$

$2 \bullet 2 \bullet 2 \bullet 3$

And in exponent form

$2^{3}\bullet 3$

Question 2

2a) Factors $30$ using factor tree

Solution:

2b) Factors of $54$ using factor tree

Solution:

2c) Factors of $28$, using tree

Solution:

$14$ is not prime, so it can be factored further.

Question 3

What is the greatest common factor between $160$ and $144$, using continuous division?

Got a number in mind you wanted to check the prime factors for? Here's a prime factor calculator you can check out.

Want to learn more related to this lesson? Take a look at how to use exponents to describe numbers, the product rule of exponents, how to find common factors of polynomials, and factoring polynomials.

Factor:

Numbers that can be multiplied together to give the original value.
Ex: The factor of 15 is 5 & 3. Because $5 \times 3 = 15$

Prime numbers:

A whole number that has only "two" factors: 1 & itself.
Ex: 2,3,5,7,11,13,17…

Composite numbers:

A Whole number that is not prime or has more than 2 factors.
Ex: 4,6,8…

Zero & One:

They are neither prime nor composite.

Factor Tree:

A method of finding prime factors by using a tree diagram.
Ex: Prime factors of $18 = 2 \times 3 \times 3$

Continuous Division:

A method of finding prime factors by using a continuous division.
Ex: Prime factors of $72 = 2 \times 2 \times 2 \times 3 \times 3$

• Determining Common Factors
• Using exponents to describe numbers

• Product rule of exponents
• Common factors of polynomials
• Factor by taking out the greatest common factor