π’ Prime Factorization
Find the prime factors of any number with full step-by-step breakdown.
How to Use This Calculator
Enter any whole number greater than 1 in the input field and click Factorize. The result shows the exponential form of the prime factorization, a check for whether the number is prime, a list of all divisors, and the total number of divisors calculated from the exponents.
Type a whole number greater than 1. The calculator handles numbers up to about 10ΒΉβ΅ using the 6kΒ±1 trial division method.
Click Factorize (or the result updates as you type).
Read the exponential form: 360 = 2Β³ Γ 3Β² Γ 5. The exponents tell you exactly how many times each prime divides the number.
Scroll down to see all divisors listed as clickable chips. Divisors are all the numbers that divide evenly into your input.
Prime Factorization Method
Every integer greater than 1 is either prime or can be written as a product of primes in exactly one way. This is the Fundamental Theorem of Arithmetic. The number of divisors comes from the exponents in the factorization: for n = pβ^a Γ pβ^b Γ pβ^c, the divisor count is (a+1)(b+1)(c+1). For 360 = 2Β³ Γ 3Β² Γ 5ΒΉ, that gives (3+1)(2+1)(1+1) = 24 divisors.
Worked Examples
Where This Comes Up in Real Life
Simplifying fractions requires finding the GCD of numerator and denominator, which requires prime factorization. To simplify 360/480, you find 360 = 2Β³ Γ 3Β² Γ 5 and 480 = 2β΅ Γ 3 Γ 5. The GCD is 2Β³ Γ 3 Γ 5 = 120. So 360/480 = 3/4. Without factorization, you might have to guess and check many numbers before finding 120.
Cryptography relies on the fact that multiplying two large prime numbers together is fast, but factoring the result back into primes is extremely slow. An RSA encryption key might be the product of two 300-digit primes. No computer on Earth can factor that product in a reasonable time. This difficulty is what makes public-key encryption secure for banking, email, and online transactions.
Frequently Asked Questions
What is prime factorization?
Prime factorization breaks a number down into its prime factors β numbers that are only divisible by 1 and themselves. Example: 60 = 2Β² Γ 3 Γ 5.
How do I find prime factors?
Divide the number by the smallest prime (2), then keep dividing the quotient by the smallest prime factor until the quotient is 1. Example: 72 β 72Γ·2=36 β 36Γ·2=18 β 18Γ·2=9 β 9Γ·3=3 β 3Γ·3=1. So 72 = 2Β³Γ3Β².
What is a prime number?
A prime number is a natural number greater than 1 that cannot be divided evenly by any number other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17...
What is the largest prime I can calculate?
This calculator handles numbers up to about 10^15 using trial division optimized with 6kΒ±1, which is fast for most practical cases.
What are divisors?
Divisors (factors) of a number are all the integers that divide it evenly. They can be derived from the prime factorization: 12 = 2Β²Γ3, divisors = 1, 2, 3, 4, 6, 12.