What is the Prime Factorization Calculator?
The Prime Factorization Calculator takes any positive integer and breaks it down into its prime factors. Simply enter a number and you'll instantly see the full prime factorization in both exponent form (e.g., 2² × 3²) and expanded form (2 × 2 × 3 × 3), along with a step-by-step factor tree. You can also find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of two numbers.
What is a Prime Number?
A prime number is a natural number greater than 1 whose only factors are 1 and itself. In other words, it cannot be formed by multiplying two smaller natural numbers. For example, 5 is prime because the only factors of 5 are 1 and 5. The number 6 is not prime because it can also be written as 2 × 3.
Key facts about prime numbers:
- There are infinitely many prime numbers.
- 2 is the only even prime number.
- 1 is not a prime number (by definition, its two "factors" are identical).
- The first 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
What is a Prime Factor?
A prime factor is a factor of a number that is itself a prime number. For example, the factors of 20 are 1, 2, 4, 5, 10, and 20. Among these, only 2 and 5 are prime — so the prime factors of 20 are 2 and 5.
What is Prime Factorization?
Prime factorization is the process of expressing a number as a product of prime numbers only. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic).
Example: 20 = 2 × 2 × 5 = 2² × 5
The factors 4 × 5 also equal 20, but 4 is not prime, so we keep dividing: 4 = 2 × 2. The final prime factorization is 2² × 5.
How to Find Prime Factorization — The Factor Tree Method
The factor tree is a visual way to find prime factors step by step:
- Choose your number (e.g., 36).
- Split it into any two factors. If the number is even, split into 2 and the other factor: 36 = 2 × 18.
- Draw two branches from 36: one to 2 (prime ✓) and one to 18.
- Repeat for 18: 18 = 2 × 9. Draw branches from 18.
- Repeat for 9: 9 = 3 × 3. Both are prime ✓.
- Collect all prime "leaves": 2, 2, 3, 3.
- Write the result: 36 = 2 × 2 × 3 × 3 = 2² × 3².
No matter how you split the tree, the final prime factorization is always the same!
Greatest Common Factor (GCF)
The Greatest Common Factor (also called GCD — Greatest Common Divisor) of two numbers is the largest number that divides both evenly. To find the GCF using prime factorization:
- Factorize both numbers.
- Find all common prime factors.
- Multiply them using the lower exponent for each shared prime.
Example: GCF(36, 60)
36 = 2² × 3²
60 = 2² × 3 × 5
Common primes: 2 (min exp = 2), 3 (min exp = 1)
GCF = 2² × 3 = 12
Least Common Multiple (LCM)
The Least Common Multiple of two numbers is the smallest number that is divisible by both. To find the LCM using prime factorization:
- Factorize both numbers.
- Take all prime factors that appear in either number.
- Use the highest exponent for each prime.
Example: LCM(36, 60)
36 = 2² × 3²
60 = 2² × 3 × 5
All primes: 2 (max exp = 2), 3 (max exp = 2), 5 (exp = 1)
LCM = 2² × 3² × 5 = 180
What is the Prime Factorization of…
| Number | Prime Factorization | Is Prime? |
|---|---|---|
| 2 | 2 | Yes |
| 12 | 2² × 3 | No |
| 36 | 2² × 3² | No |
| 100 | 2² × 5² | No |
| 360 | 2³ × 3² × 5 | No |
| 1024 | 2¹⁰ | No |
| 9973 | 9973 | Yes |