Direct answer: Prime factorization means breaking a number into a product of prime numbers — numbers divisible only by 1 and themselves. To find it, divide by the smallest prime that divides evenly, repeat with the quotient until only primes remain, then multiply them together. For 60, that gives 2² × 3 × 5, using either the factor tree or the division (ladder) method.
Prime factorization is the skill underneath half of middle-school math: it is how you find the GCF and LCM, simplify fractions, and count how many factors a number has. This guide teaches both methods — the factor tree and the division method — side by side, so you can pick the one that fits the problem in front of you.
If you want to check your work, here is how to find factors of a number instantly — Calcrio’s free tool gives the prime factorization of any number, along with its full factor list, factor pairs, GCF and LCM.
What Is Prime Factorization?
Prime factorization is writing a number as a product of prime numbers only. A prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, and 19.
A few examples:
- 12 = 2 × 2 × 3 = 2² × 3
- 36 = 2 × 2 × 3 × 3 = 2² × 3²
- 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Compare that with ordinary factoring, which just splits a number into any two factors — 36 = 4 × 9, for instance. Prime factorization keeps splitting until nothing is left but primes. That is the difference between factoring and prime factorization.
One more fact: every whole number greater than 1 has exactly one prime factorization. It does not matter which pair you start with or which method you use — the primes are always the same. That means there is always one right answer to check against.
How to Find Prime Factorization: The Division Method
The division method — sometimes called the ladder method — is the most reliable way to find prime factorization, especially for larger numbers:
- Write the number with a short vertical line to its left, like a ladder.
- Divide by the smallest prime that divides it evenly. Start with 2, then try 3, 5, 7, and so on.
- Write the quotient below, then repeat with the new number.
- Stop when the quotient is 1. The primes you divided by, read top to bottom, are the prime factorization.
Worked Example: The Prime Factorization of 60
The smallest prime dividing 60 is 2: 60 ÷ 2 = 30. Divide 30 by 2: 30 ÷ 2 = 15. Next, 2 does not divide 15, so move to 3: 15 ÷ 3 = 5. Finally, 5 ÷ 5 = 1, and we stop.
The primes used, in order, are 2, 2, 3, and 5.
The prime factorization of 60 is 2 × 2 × 3 × 5 = 2² × 3 × 5.
Check it: 2 × 2 = 4, 4 × 3 = 12, 12 × 5 = 60. Correct.
Worked Example: The Prime Factorization of 72
72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1. The primes used are 2, 2, 2, 3, 3.
The prime factorization of 72 is 2 × 2 × 2 × 3 × 3 = 2³ × 3².
Check it: 2 × 2 × 2 = 8, 3 × 3 = 9, and 8 × 9 = 72. Correct.
Worked Example: The Prime Factorization of 100
100 ÷ 2 = 50, 50 ÷ 2 = 25, 25 ÷ 5 = 5, 5 ÷ 5 = 1. The primes used are 2, 2, 5, 5.
The prime factorization of 100 is 2 × 2 × 5 × 5 = 2² × 5².
Check it: 2 × 2 = 4, 5 × 5 = 25, and 4 × 25 = 100. Correct.
Notice the pattern: you always try 2 first, and you only move to a bigger prime when the current one stops working. That discipline is what makes the division method nearly error-proof — and it is why teachers love it for exams.
How to Find Prime Factorization: The Factor Tree Method
The factor tree is the visual method: split the number into two factors, then split each factor again, branch by branch, until every branch ends in a prime number. The primes at the ends — the “leaves” — are the prime factorization.
Here is the method, step by step:
- Write the number at the top.
- Split it into any two factors and write them on branches below.
- Circle any prime numbers — they are finished.
- Split any composite number again, and keep going.
- Collect the circled primes at the ends of the branches and multiply them together.
Worked Example: The Factor Tree of 60
Split 60 into 6 × 10. Then split 6 into 2 × 3 and 10 into 2 × 5 — all prime, so circle them. The leaves are 2, 3, 2, and 5.
60 = 2² × 3 × 5 — the same answer the division method gave. Any starting pair gives the same primes.
For a full visual walk-through, see our guide to what a factor tree is and how to draw one.
Factor Tree vs Division Method: Which Should You Use?
Most guides teach one method. The table below compares both, so you can choose per problem.
| Factor tree | Division (ladder) method | |
|---|---|---|
| Best for | Smaller numbers, visual learners, homework | Larger numbers, exams, any size |
| How it works | Split into factor pairs, branch by branch | Divide by the smallest prime, step by step |
| Speed | Fast for numbers under 100 | Faster for numbers over 100 |
| Error risk | Medium — easy to miss a branch | Low — the ladder keeps you on track |
| Shows the factors | Yes — every split is visible | No — only the primes |
| Best for GCF/LCM | Good, with the tree drawn out | Better — primes line up in a column |
| Example | 72 → 8 × 9 → 2³ × 3² | 72 ÷ 2 ÷ 2 ÷ 2 ÷ 3 ÷ 3 = 1 |
Learn both, then use the division method when the number is large or the clock is ticking, and the factor tree when you want to see the structure or explain it to someone else. On a test, the division method wins — it is compact, and the ladder format makes it easy to check each step.
A Bigger Worked Example: The Prime Factorization of 360
For a larger number, the division method shows its strength:
360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1.
The primes used are 2, 2, 2, 3, 3, 5.
The prime factorization of 360 is 2³ × 3² × 5.
This one is worth remembering: it tells you how many factors 360 has. Add 1 to each exponent and multiply: (3 + 1) × (2 + 1) × (1 + 1) = 4 × 3 × 2 = 24. So 360 has 24 factors — a formula that works only because prime factorization is unique.
Why Is Prime Factorization Important?
Prime factorization is the engine behind three skills you will use constantly:
- Finding the GCF and LCM. Break both numbers into primes, take the lowest powers of the shared primes for the GCF and the highest powers for the LCM. For 12 and 18, the GCF is 2 × 3 = 6 and the LCM is 2² × 3² = 36. Our GCF vs LCM guide works through this in full.
- Simplifying fractions. To reduce 18/24 to 3/4, divide top and bottom by their GCF, 6 — which comes from prime factorization.
- Counting factors. The exponents in the prime factorization tell you exactly how many factors a number has — as shown with 360 above.
It also has a famous real-world use: the encryption behind online banking relies on the fact that factoring very large numbers into primes is extremely hard for computers. Prime factorization is easy for small numbers and nearly impossible for huge ones — and that gap keeps the internet secure.
3 Common Mistakes in Prime Factorization
Mistake 1: Stopping too early. 36 = 4 × 9 is factoring, not prime factorization. Keep splitting until every factor is prime: 36 = 2² × 3². If any factor in your answer is composite, you are not done.
Mistake 2: Forgetting small primes first. Dividing 60 by 3 first (60 ÷ 3 = 20) still reaches the right answer — but you risk missing a factor of 2 later. Always start with 2 and work upward.
Mistake 3: Adding instead of multiplying. The prime factors of 60 are 2, 2, 3, and 5. Their product is 60, not their sum (2 + 2 + 3 + 5 = 12). Always verify by multiplying your primes back together.
Check Your Answer with Calcrio’s Free Tool
The free tool at Calcrio finds the prime factorization of any number, plus its complete list of factors of a number, factor pairs, and the GCF and LCM of two numbers. It is free and instant. But the method matters more than the tool: once you can run the division ladder by hand, prime factorization becomes a two-minute skill.
If you have not yet learned the basics, start with how to find factors of a number — the step-by-step method and square-root shortcut that make all of this faster.
Frequently Asked Questions
What is prime factorization?
Prime factorization is writing a number as a product of prime numbers only. For example, 60 = 2² × 3 × 5, because 2, 3, and 5 are all prime and their product is 60. Every number greater than 1 has exactly one prime factorization.
How do you find the prime factorization of a number?
Divide the number by the smallest prime that divides it evenly, repeat with each quotient, and stop when the quotient is 1. The primes you divided by are the prime factorization. For 72: 72 ÷ 2 = 36, ÷ 2 = 18, ÷ 2 = 9, ÷ 3 = 3, ÷ 3 = 1, so 72 = 2³ × 3².
What is the difference between factoring and prime factorization?
Factoring splits a number into any two factors, like 36 = 4 × 9. Prime factorization keeps splitting until every factor is prime, like 36 = 2² × 3². Factoring is just the first step.
What is the prime factorization of 100?
The prime factorization of 100 is 2² × 5². Working it out: 100 ÷ 2 = 50, ÷ 2 = 25, ÷ 5 = 5, ÷ 5 = 1, so the primes used are 2, 2, 5, and 5. Check: 4 × 25 = 100.
What is the prime factorization of 72?
The prime factorization of 72 is 2³ × 3². Working it out: 72 ÷ 2 = 36, ÷ 2 = 18, ÷ 2 = 9, ÷ 3 = 3, ÷ 3 = 1. Check: 8 × 9 = 72.
Why is prime factorization important?
Prime factorization is how you find the GCF and LCM, simplify fractions, and count how many factors a number has. It also powers modern encryption: factoring very large numbers into primes is so hard for computers that it keeps online banking secure.