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Prime factorization breaks a whole number into the prime numbers that multiply to make it — the number's unique "fingerprint." Every integer greater than 1 has exactly one prime factorization (the Fundamental Theorem of Arithmetic), and it underpins simplifying fractions, finding the GCF and LCM, and much of number theory and cryptography.
Enter a number and its prime factorization is calculated instantly.
How is it calculated?
What it means
A prime number has exactly two divisors: 1 and itself (2, 3, 5, 7, 11…). Prime factorization writes a number as a product of primes: 360 = 2³ × 3² × 5.
How it's found
Divide by the smallest prime that fits, repeatedly, until you reach 1: - 360 ÷ 2 = 180 ÷ 2 = 90 ÷ 2 = 45 → three 2s - 45 ÷ 3 = 15 ÷ 3 = 5 → two 3s - 5 ÷ 5 = 1 → one 5
So 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Why it's useful
- Simplifying fractions: shared prime factors cancel.
- GCF and LCM: the greatest common factor uses the smallest shared exponents, the least common multiple the largest — a GCF/LCM calculator builds directly on this.
- Divisor count: from the exponents, the number of divisors is (3+1)(2+1)(1+1) = 24 for 360.
- Cryptography: the difficulty of factoring very large numbers underpins modern encryption.
Uniqueness
By the Fundamental Theorem of Arithmetic, every integer above 1 has one and only one prime factorization (ignoring order). That uniqueness is what makes it a reliable fingerprint.
Worked example
Take 360: divide by 2 three times (360 → 180 → 90 → 45), then by 3 twice (45 → 15 → 5), then by 5 once (→ 1). The result is 360 = 2³ × 3² × 5. From this you can read off a lot: the number of divisors is (3+1) × (2+1) × (1+1) = 24, and to find the GCF or LCM with another number you just compare the exponents of each prime. For 84 = 2² × 3 × 7, the shared primes with 360 are 2² and 3, so their GCF is 2² × 3 = 12.
FAQ
How is prime factorization calculated?+
Divide the number by the smallest prime that fits, repeatedly, until you reach 1. The primes you divided by, with their counts, are the factorization: 360 = 2³ × 3² × 5.
What is a prime number?+
A number greater than 1 with exactly two divisors, 1 and itself: 2, 3, 5, 7, 11, 13… 2 is the only even prime. 1 is not prime.
Does every number have a unique prime factorization?+
Yes — the Fundamental Theorem of Arithmetic guarantees every integer above 1 has exactly one prime factorization (ignoring the order of factors).
How does prime factorization help with fractions?+
Factor the numerator and denominator; shared prime factors cancel, leaving the fraction in lowest terms. It's the systematic way to simplify.
How do I find the number of divisors from the factorization?+
Add 1 to each exponent and multiply: 360 = 2³ × 3² × 5 has (3+1)(2+1)(1+1) = 24 divisors.
Why is prime factorization important in cryptography?+
Multiplying two large primes is easy, but factoring the result back is extremely hard. That asymmetry is the basis of widely used public-key encryption.