Break any number into its prime factors instantly. Find prime factorization using trial division or factor tree methods with step-by-step solutions. Free online prime factorization calculator for students and math enthusiasts.
Please provide an integer to find its prime factors as well as a factor tree.
Prime numbers are natural numbers greater than 1 that cannot be formed by multiplying two smaller numbers. An example is 7, which can only be formed by 1 × 7. Other examples: 2, 3, 5, 11, etc. Numbers that can be formed with two other numbers greater than 1 are called composite numbers (4, 6, 9, etc.).
The fundamental theorem of arithmetic states that every natural number greater than 1 is either prime or can be factored as a unique product of prime numbers. Example: 60 = 5 × 3 × 2 × 2.
| Number | Prime Factorization |
|---|---|
| 2 | 2 |
| 3 | 3 |
| 4 | 2² |
| 5 | 5 |
| 6 | 2 × 3 |
| 7 | 7 |
| 8 | 2³ |
| 9 | 3² |
| 10 | 2 × 5 |
| 12 | 2² × 3 |
| 15 | 3 × 5 |
| 18 | 2 × 3² |
| 20 | 2² × 5 |
| 24 | 2³ × 3 |
| 30 | 2 × 3 × 5 |
| 36 | 2² × 3² |
| 48 | 2⁴ × 3 |
| 60 | 2² × 3 × 5 |
| 72 | 2³ × 3² |
| 100 | 2² × 5² |
Let's start simple. Prime factorization is just a fancy way of saying "break a number down into the smallest building blocks possible."
You know how LEGOs come in different shapes and sizes? Well, numbers are kind of like that. Some numbers are "prime" - they can't be broken down any further. Like 2, 3, 5, 7, 11, 13. These are the basic LEGO bricks of math.
So when you do prime factorization, you're asking: "What prime numbers do I need to multiply together to get this number?"
For example, the prime factorization of 12 is 2 × 2 × 3. Because 2 × 2 × 3 = 12, and 2 and 3 are both prime numbers.
Sounds simple, right? It is. But it gets trickier with bigger numbers.
Using our calculator is stupidly easy. Here's how:
But wait - there's more. Our calculator doesn't just show the final answer. It shows you each division step along the way. So you can actually learn how it's done.
Let me show you an example. Say you type in 72. Here's what you'll see:
Step 1: 72 ÷ 2 = 36
Step 2: 36 ÷ 2 = 18
Step 3: 18 ÷ 2 = 9
Step 4: 9 ÷ 3 = 3
Step 5: 3 ÷ 3 = 1
So the prime factors of 72 are: 2 × 2 × 2 × 3 × 3
Or written with exponents: 2³ × 3²
See how that works? Each time we divide by the smallest prime number possible until we get to 1.
Some people learn better with pictures. That's where the factor tree comes in.
Instead of writing out division steps, you draw a tree. Start with your number at the top. Then branch it into two factors. Keep branching until all the "leaves" are prime numbers.
Let's do 72 again with a factor tree:
Start: 72
Branch: 72 = 8 × 9
Branch 8: 8 = 2 × 4
Branch 4: 4 = 2 × 2
Branch 9: 9 = 3 × 3
Now collect all the prime "leaves": 2, 2, 2, 3, 3
Same answer! 2 × 2 × 2 × 3 × 3 = 72
Our calculator doesn't draw the tree for you (yet), but you can use the step-by-step output to build your own tree. It's a great way to check your homework.
You might be thinking: "When am I ever going to use this?" Fair question. Here are some real-world uses:
I've seen students make the same mistakes over and over. Here's what to watch out for:
Mistake #1: Forgetting That 1 Is NOT a Prime Number
This is the biggest one. 1 is not prime. A prime number has exactly two factors: 1 and itself. 1 only has one factor (itself), so it doesn't count. Don't include 1 in your prime factors.
Mistake #2: Stopping Too Early
You're not done until every factor is prime. If you have a 4 or a 9 or a 15 in your list, keep going. Those aren't prime.
Mistake #3: Forgetting to Check All Prime Numbers
When you're dividing, don't just try 2 and 3. You might need 5, 7, 11, 13, or bigger primes. For example, the prime factors of 143 are 11 and 13. If you stop at 7, you'll miss them.
Mistake #4: Mixing Up Factors and Multiples
Factors are numbers that divide evenly into your number. Multiples are numbers you get by multiplying your number by something. Don't confuse them.
Here's a simple trick: multiply all your prime factors together. If you get back to your original number, you did it right.
Let's test with 72: 2 × 2 × 2 × 3 × 3 = 8 × 9 = 72. Perfect.
If you get a different number, something went wrong. Go back and check each step.
What happens if you type in a prime number, like 17 or 23? Our calculator will tell you that the number is already prime. Its only prime factor is itself. So the prime factorization of 17 is just 17. Simple as that.
Start with the smallest primes first. Always try 2 first, then 3, then 5, then 7. Work your way up.
Use divisibility rules. If a number ends in 0, 2, 4, 6, or 8, it's divisible by 2. If the digits add up to a multiple of 3, it's divisible by 3. If it ends in 0 or 5, it's divisible by 5.
Write down each step. Don't try to do it all in your head. Writing it out helps you catch mistakes.
Practice with small numbers first. Start with numbers under 100, then work your way up to bigger ones.
Use our calculator to check your homework. Do the problem yourself first, then use the calculator to see if you got it right.
Once you get comfortable, you'll want to write your answer using exponents. Instead of writing 2 × 2 × 2 × 3 × 3, you write 2³ × 3². It's shorter and cleaner. Our calculator shows both formats, so you can learn how to convert between them.
The Fundamental Theorem of Arithmetic says that every number has exactly one unique set of prime factors (if you ignore the order). This is a huge deal in math.
The largest known prime number has over 24 million digits. Good luck factoring that one.
Ancient Greek mathematicians like Euclid studied prime numbers over 2,000 years ago.
Prime numbers are used in RSA encryption, which protects your online banking and shopping.
Problem: The calculator says "Invalid input."
Solution: Make sure you're typing a whole number. No decimals, no fractions, no letters.
Problem: The answer looks too long.
Solution: That's normal for big numbers. Just check that all the factors are prime.
Problem: I don't understand the steps.
Solution: Read through each division step slowly. Each time, we're dividing by the smallest prime number that goes evenly into the result.
No, 1 is not a prime number. A prime number has exactly two factors: 1 and itself. 1 only has one factor, so it doesn't qualify. This is a super common mistake, so don't feel bad if you thought it was prime.
Start by trying the smallest prime numbers: 2, 3, 5, 7, 11, 13. For 1001, you'll find it's divisible by 7 (1001 ÷ 7 = 143). Then factor 143, which is 11 × 13. So the prime factors of 1001 are 7 × 11 × 13. Our calculator can do this for you in seconds.
Regular factors are any numbers that divide evenly into your number. For 12, the factors are 1, 2, 3, 4, 6, and 12. Prime factors are only the prime numbers among those factors. For 12, the prime factors are 2 and 3 (since 2 × 2 × 3 = 12).
You're done when every factor in your list is a prime number. If you see a 4, 6, 8, 9, 10, or any composite number, you need to keep breaking it down. The final answer should only contain prime numbers.
That depends on your teacher. Some allow calculators, some don't. But even if you can't use one during the test, our calculator is great for practicing at home. Do a few problems on your own, then check your answers here.
It's used in cryptography (keeping your online info safe), simplifying fractions, finding GCF and LCM, and understanding number patterns. Plus, it's a foundational skill for more advanced math like algebra and number theory.
Then you're done! The prime factorization of a prime number is just itself. For example, the prime factorization of 17 is 17. Our calculator will tell you this.
Instead of writing 2 × 2 × 2 × 3 × 3, count how many times each prime appears. 2 appears three times, so write 2³. 3 appears twice, so write 3². The final answer is 2³ × 3². Our calculator shows both formats.
Start with the smallest prime (2) and work your way up. Use divisibility rules to check quickly. Write down each division step. And practice - the more you do it, the faster you'll get. Our calculator can help you check your speed and accuracy.
Prime factorization is only for positive whole numbers (integers greater than 1). Negative numbers and decimals don't have prime factors in the traditional sense. If you need to factor a negative number, just factor the positive version and add a negative sign.