Prime Factorization Calculator
Enter any number to find its complete prime factorization. Shows the step-by-step division process, exponent form, and total number of divisors.
How Prime Factorization Works
Prime factorization is the process of breaking a number down into a product of prime numbers. Every integer greater than 1 has exactly one prime factorization — this is called the Fundamental Theorem of Arithmetic.
A prime number is a number greater than 1 that can only be divided evenly by 1 and itself (2, 3, 5, 7, 11, 13, …). A composite number can be divided by at least one other positive integer besides 1 and itself.
The Trial Division Method
The standard algorithm for small numbers:
- Start with the smallest prime: 2
- If the number divides evenly, record the prime and divide
- Repeat until the prime no longer divides evenly
- Move to the next prime (3, 5, 7, 11, …)
- Stop when the quotient reaches 1
Example: Prime Factorization of 360
360 ÷ 2 = 180 180 ÷ 2 = 90 90 ÷ 2 = 45 45 ÷ 3 = 15 15 ÷ 3 = 5 5 ÷ 5 = 1 360 = 2³ × 3² × 5
Example: Prime Factorization of 84
84 ÷ 2 = 42 42 ÷ 2 = 21 21 ÷ 3 = 7 7 ÷ 7 = 1 84 = 2² × 3 × 7
Common Prime Factorizations
Quick reference for frequently searched numbers:
| Number | Prime Factorization | Prime Factors | Divisors |
|---|---|---|---|
| 12 | 2² × 3 | 2, 3 | 6 |
| 24 | 2³ × 3 | 2, 3 | 8 |
| 36 | 2² × 3² | 2, 3 | 9 |
| 48 | 2⁴ × 3 | 2, 3 | 10 |
| 60 | 2² × 3 × 5 | 2, 3, 5 | 12 |
| 72 | 2³ × 3² | 2, 3 | 12 |
| 100 | 2² × 5² | 2, 5 | 9 |
| 120 | 2³ × 3 × 5 | 2, 3, 5 | 16 |
| 144 | 2⁴ × 3² | 2, 3 | 15 |
| 180 | 2² × 3² × 5 | 2, 3, 5 | 18 |
| 200 | 2³ × 5² | 2, 5 | 12 |
| 256 | 2⁸ | 2 | 9 |
| 360 | 2³ × 3² × 5 | 2, 3, 5 | 24 |
| 500 | 2² × 5³ | 2, 5 | 12 |
| 1000 | 2³ × 5³ | 2, 5 | 16 |
Counting Divisors from Prime Factorization
Once you have the prime factorization, you can calculate the total number of divisors without listing them all.
If a number's factorization is p₁a₁ × p₂a₂ × … × pₖaₖ, the number of divisors is:
Number of divisors = (a₁ + 1) × (a₂ + 1) × … × (aₖ + 1)
Worked Examples
How many divisors does 60 have?
60 = 2² × 3¹ × 5¹ Divisors = (2+1)(1+1)(1+1) = 3 × 2 × 2 = 12 All 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
How many divisors does 1000 have?
1000 = 2³ × 5³ Divisors = (3+1)(3+1) = 4 × 4 = 16 All 16 divisors: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
Why Prime Factorization Matters
Simplifying Fractions
To simplify a fraction, find the prime factorization of both numerator and denominator, then cancel common factors.
Simplify 84/120: 84 = 2² × 3 × 7 120 = 2³ × 3 × 5 Cancel common factors (2² × 3): 84/120 = 7/10
Finding GCD and LCM
GCD (Greatest Common Divisor): multiply the lowest powers of all shared primes.
LCM (Least Common Multiple): multiply the highest powers of all primes present in either number.
GCD and LCM of 36 and 60: 36 = 2² × 3² 60 = 2² × 3 × 5 GCD = 2² × 3 = 12 (lowest shared powers) LCM = 2² × 3² × 5 = 180 (highest powers of all primes)
Cryptography
RSA encryption — used to secure websites, emails, and digital signatures — relies on the fact that multiplying two large primes is easy, but factoring the product back into those primes is extremely hard. A 2048-bit RSA key involves primes with over 300 digits each.
Frequently Asked Questions
What is prime factorization?
Prime factorization is the process of expressing a number as a product of prime numbers. Every integer greater than 1 has a unique prime factorization. For example, 60 = 2² × 3 × 5. This is guaranteed by the Fundamental Theorem of Arithmetic.
How do you find the prime factorization of a number?
Divide the number by the smallest prime (2) repeatedly until it no longer divides evenly. Then try 3, then 5, and so on. Continue until the quotient is 1. Example: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, 7 ÷ 7 = 1. So 84 = 2² × 3 × 7.
What is the prime factorization of 100?
The prime factorization of 100 is 2² × 5² (2 × 2 × 5 × 5 = 100). The prime factors are 2 and 5.
What is the prime factorization of 72?
The prime factorization of 72 is 2³ × 3² (2 × 2 × 2 × 3 × 3 = 72). The prime factors are 2 and 3.
Is 1 a prime number?
No. A prime number must be greater than 1. The number 1 is neither prime nor composite. It is excluded from prime factorization by definition.
What is the Fundamental Theorem of Arithmetic?
It states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of factors. For example, 30 can only be written as 2 × 3 × 5 — no other set of primes multiplies to 30.
How many divisors does a number have?
If a number's prime factorization is p₁a₁ × p₂a₂ × … × pₖaₖ, the total number of divisors is (a₁ + 1)(a₂ + 1)…(aₖ + 1). For example, 60 = 2² × 3 × 5 has (2+1)(1+1)(1+1) = 12 divisors.
What is the difference between prime factors and all factors?
Prime factors are only the prime numbers in the factorization. All factors (divisors) include every number that divides evenly into the original number. For 12: prime factors are 2 and 3, but all divisors are 1, 2, 3, 4, 6, and 12.
Why is prime factorization useful?
It is used to simplify fractions, find GCD and LCM, count the total number of divisors, and is foundational in cryptography (RSA encryption relies on the difficulty of factoring large numbers).
Can prime factorization be applied to negative numbers?
Prime factorization is defined for positive integers greater than 1. For negative numbers, factor out −1 first, then factorize the absolute value. For example, −60 = −1 × 2² × 3 × 5.
Related Tools
Explore these related calculators for more number theory and math operations:
- Prime Number Checker - Test if a number is prime
- GCD LCM Calculator - Find greatest common divisor and least common multiple
- Factoring Calculator - Find all factors of any number
- Fraction Calculator - Simplify fractions using prime factorization
- Exponent Calculator - Calculate powers and exponents
Privacy & Limitations
- All calculations run entirely in your browser -- nothing is sent to any server.
- Results are computed using standard formulas and should be verified for critical applications.
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Prime Factorization Calculator FAQ
What is prime factorization?
Prime factorization is the process of expressing a number as a product of prime numbers. Every integer greater than 1 has a unique prime factorization. For example, 60 = 2² × 3 × 5. This is guaranteed by the Fundamental Theorem of Arithmetic.
How do you find the prime factorization of a number?
Divide the number by the smallest prime (2) repeatedly until it no longer divides evenly. Then try 3, then 5, and so on. Continue until the quotient is 1. Example: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, 7 ÷ 7 = 1. So 84 = 2² × 3 × 7.
What is the prime factorization of 100?
The prime factorization of 100 is 2² × 5² (2 × 2 × 5 × 5 = 100). The prime factors are 2 and 5.
What is the prime factorization of 72?
The prime factorization of 72 is 2³ × 3² (2 × 2 × 2 × 3 × 3 = 72). The prime factors are 2 and 3.
Is 1 a prime number?
No. By definition, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The number 1 is neither prime nor composite. It is excluded from prime factorization.
What is the Fundamental Theorem of Arithmetic?
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. For example, 30 can only be written as 2 × 3 × 5 — no other combination of primes produces 30.
How many divisors does a number have?
If a number's prime factorization is p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, then the total number of divisors is (a₁ + 1)(a₂ + 1)...(aₖ + 1). For example, 60 = 2² × 3 × 5 has (2+1)(1+1)(1+1) = 12 divisors.
What is the difference between prime factors and all factors?
Prime factors are only the prime numbers in the factorization. All factors (divisors) include every number that divides evenly into the original number. For 12: prime factors are 2 and 3, but all factors are 1, 2, 3, 4, 6, and 12.
Why is prime factorization useful?
Prime factorization is used to simplify fractions, find GCD and LCM, determine the number of divisors, solve problems in cryptography (RSA encryption relies on the difficulty of factoring large numbers), and is foundational in number theory.
Can prime factorization be applied to negative numbers?
Prime factorization is defined for positive integers greater than 1. For negative numbers, you can factor out -1 and then find the prime factorization of the absolute value. For example, -60 = -1 × 2² × 3 × 5.