What Is Prime Factorization? Explain It to a 10-Year-Old

Jan 30, 2025 | Queen Creek
Mathnasium math tutor solving a math problem with a student.

Prime factorization is like discovering the secret ingredients that create a number!

In this guide, we’ll explore what prime factorization is, how it works, and why it’s so useful—all explained in a way that’s easy to understand. 

Ready to uncover the magic of numbers? Let’s dive in!


What is Prime Factorization?

Prime factorization is the process of breaking down a number into its smallest building blocks: prime numbers.

In this process, we discover how prime numbers multiply together to create our original number.

Not sure what prime numbers are?

We’ve got you covered! 

Let’s brush up on some of the key terms in prime factorization:


Prime Numbers

A prime number is a number that has exactly two factors: 1 and itself. 

Numbers 2, 3, 5, 7, and 11 are prime numbers because you can’t divide them by anything other than 1 and themselves without leaving a remainder.

Prime numbers are like the “atoms” of math—you can’t break them down any further!

Fun Fact: The smallest prime number is 2, and it’s the only even prime number. Why? Because all other even numbers are divisible by 2, so they’re not prime numbers!


Prime Factors

A factor is a number that divides another number evenly without leaving a remainder. For example, the number 12 has six factors: 1, 2, 3, 4, 6, and 12.

Prime factors are the prime numbers that multiply together to make up a given number, hence the name of our process: prime factorization. For example:

  • The prime factors of 12 (given number) are 2 × 2 × 3 (or 22 × 3).

  • The prime factors of 18 are 2 × 3 × 3 (or 2 × 32).


How to Do Prime Factorization?

Prime factorization is like solving a puzzle where you break a number into its simplest building blocks—prime numbers!

There are a few ways in which we can solve this puzzle.

We have a fun video guide to prime factorization that you may want to check out!

Or, read on and let’s discover the steps together!

Let’s look at the different methods to prime factorization. 

Give them all a try to find out which one you like best.


1. Factor Tree Method

The Factor Tree Method is a fun and visual way to break a number into its prime factors.

Imagine the number as the "trunk" of a tree and the factors as the "branches." We keep splitting the branches until all the "leaves" are prime numbers. It’s like growing a tree of math facts!

If we are to go step-by-step:

  1. Start with the number you want to factorize.

  2. Split the number into any two factors.

  3. Check if the factors are prime. If not, break them down further into factors.

  4. Repeat until all branches end in prime numbers.

  5. Write the original number as a product of these primes.


Let’s try using the Factor Tree Method for prime factorization of the number 36.

  1. Our number is 36

  2. Now we can split 36 into any two factors. Let’s say 6 and 6.

  3. The factors are not prime numbers, which means we have to break them down further:

6 = 2 × 3 and 6 = 2 × 3

  1. Now we have our prime factors of 36: 2 × 2 × 3 × 3

  2. Finally, we can write 36 as:

36 = 2 × 2 × 3 × 3 or 36 = 22 × 32

Illustration of the factor tree method for prime factorization.


2. Division Method

The Division Method is a step-by-step process of dividing a number by the smallest prime numbers until you can’t divide anymore.

It’s like peeling layers off an onion, one prime at a time, until you’re left with only prime factors.

The steps are:

  1. Start dividing the number by the smallest prime (2, 3, 5, etc.) that divides evenly.

  2. Continue dividing the quotient by the smallest prime until the quotient becomes 1.

  3. The prime factors are the divisors used.


Let’s see how the Division Method works when doing the prime factorization for the number 84:

  1. First, we divide 84 by the smallest prime factor:

84 ÷ 2 = 42

  1. The quotient is 42 and not 1, so we need to divide it further by the smallest prime factors available until we reach 1:

42 ÷ 2 = 21

21 ÷ 3 = 7

7 ÷ 7 = 1

  1. Now that we’ve reached 1, we can write 84 as the product of the prime numbers we used when dividing:

84 = 2 × 2 × 3 × 7 

or 

84 = 22 × 3 × 7

Illustration of the division method for prime factorization.


3. Exponentiation

The Exponentiation Method is a way to simplify prime factorization by grouping identical prime factors and expressing them using exponents. 

Instead of listing the same prime numbers multiple times, you can represent them compactly with a power.

We used this in in the Division Method where we simplified 2 × 2 using the exponent 22

84 = 2 × 2 × 3 × 7  

84 = 22 × 3 × 7.

The steps include:

  1. Perform factorization by using a factor tree or division method.

  2. Group identical prime factors and express them using exponents.


Example: Factorize 48.

  1. First, we factorize 48:

48 = 2 × 24 = 2 × (2 × 12) = 2 × (2 × (2 × 6)) = 2 × (2 × (2 × (2 × 3)))

  1. Now we can group all the 2’s together. We have four 2’s, which means that 2 was multiplied by itself four times. Don’t forget to include the 3 in the answer!

Final answer: 24 × 3


Want to brush up on exponents? Check out our fun video guide:


When Do We Use Prime Factorization in Math?

Prime factorization has many applications in mathematics. 

Understanding how numbers break into prime factors is a valuable tool for simplifying fractions and even solving real-life problems. 

Let’s explore two common uses: finding the Greatest Common Factor (GCF) and the Least Common Multiple (LCM).


Greatest Common Factor

The Greatest Common Factor (GCF), also called the Highest Common Factor (HCF), is the largest number that can divide two or more numbers evenly. 

Prime factorization makes finding the GCF easy by identifying the prime factors that the numbers share.


How To Find The GCF

To find the GCF, we should first perform prime factorization on each number. Then, we identify the prime factors common to all numbers. Finally, we multiply the common prime factors and get the GCF!


Let’s practice using prime factorization to find the GCF of 24 and 36:

First, we perform prime factorization on each number:

  • Prime factorization of 24: 2 × 2 × 2 × 3 = 23 × 3

  • Prime factorization of 36: 2 × 2 × 3 × 3 = 22 × 32

Now, we find the prime factors common to both 24 and 36, which will be 2, 2, and 3. Finally, we multiply them and get our answer!

  • Common prime factors: 2 × 2 × 3 = 12

  • Final Answer: GCF = 12


Least Common Multiple

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. 

Using prime factorization, we can find the LCM by combining all prime factors, taking the highest power of each factor.


How To Find The LCM

Similarly to finding the HCF, we first perform prime factorization on each number. It is best to use exponentiation to make things easier. 

This is where exponentiation comes in handy–we then list all the prime factors by using the highest power of each prime. Lastly, we should multiply these prime factors to get our LCM.


Let’s practice using prime factorization to find the LCM of 24 and 36:

First, we perform prime factorization on each number:

  • Prime factorization of 24: 2 × 2 × 2 × 3 = 23 × 3

  • Prime factorization of 36: 2 × 2 × 3 × 3 = 22 × 32

Now we take the highest power of each prime: 23 and 32

Finally, we should multiply 23 and 32:

2 × 3 × 3 × 2 = 72

Final Answer: LCM = 72


Need to refresh on the GCD and LCM? Check out our tutorial:


Practice Prime Factorization!

Find the prime factorization for these five numbers:

  1. 45

  2. 38

  3. 126

  4. 299

  5. 250

Scroll to the bottom of the page to find out the answers!


FAQs about Prime Factorization

At Mathnasium of Queen Creek, we work with students of all skill levels, empowering them to learn and master K-12 math concepts such as prime factorization. 

Let’s look at some of the questions our students asked during their sessions:


1. How do I know when to stop breaking the number into factors?

You stop breaking down the numbers when all the factors are prime numbers. 

Why? 

Prime numbers can’t be broken down any further.


2. Why is 1 not a prime number?

For a number to be considered prime, it needs exactly two factors: 1 and itself. 

Since 1 only has one factor (itself), it’s not considered prime.


3. What’s the difference between a factor tree and division?

Both a factor tree and division method are tools for finding the prime factorization of a number, but they differ in how the process is organized. 

A factor tree uses branching to show the process, which may be appealing to visual learners, while the division method uses repeated division to list the prime factors step by step.


4. Why is prime factorization useful?

Prime factorization helps us solve problems in math like finding the greatest common factor (GCF), the least common multiple (LCM), and even simplifying fractions!


Master Prime Factorization with Top-Rated Math Tutors at Mathnasium of Queen Creek

Mathnasium of Queen Creek is a math-only learning center in Queen Creek, AZ.

Our specially trained math tutors work with elementary school students of all skill levels to help them understand and excel in any math class and topic, including prime factorization.

Explore our approach to elementary school tutoring:


Students begin their Mathnasium journey with a free diagnostic assessment that helps us understand their learning style, strengths, and areas for improvement. Using assessment-based insights, our specially trained tutors create personalized learning plans that will put the student on their best path to math mastery.

Whether they are looking to catch up, keep up, or get ahead on their math journey, Mathnasium of Queen Creek empowers students to unlock their math potential.

Schedule a free assessment and enroll today! 



Psst! Check Your Answers

  1. 3 × 3 × 5 or 32 × 5

  2. 2 × 19

  3. 2 × 3 × 3 × 7 or 2 × 32 × 7

  4. 13 × 23

  5. 2 × 5 × 5 × 5 or 2 × 53