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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.
Prime Factorization Calculator: step-by-step worked example
Prime factorization of 360 (division ladder).
- Number
- 360
- 1
Divide by the smallest prime repeatedly
n ÷ p while p divides n, then move to the next prime
360 ÷ 2 = 180 · 180 ÷ 2 = 90 · 90 ÷ 2 = 45 · 45 ÷ 3 = 15 · 15 ÷ 3 = 5 · 5 ÷ 5 = 1 = 2, 2, 2, 3, 3, 5
- 2
Group equal factors as powers
n = p₁ᵉ¹ × p₂ᵉ² × …
360 = 2³ × 3² × 5 = 2^3 × 3^2 × 5
- 3
Distinct prime factors
How many different p?
= 3
- 4
Number of positive divisors
(e₁ + 1) × (e₂ + 1) × …
(3 + 1) × (2 + 1) × (1 + 1) = 24
- 23
- 32
- 51
Good to know: You only need to test divisors up to √n — whatever is left above 1 is itself prime. 1 is not prime; 2 is the only even prime. Factorizations are the basis of GCF/LCM and simplifying roots.
Prime Factorization Calculator — how it works
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.
Prime factorisation of common numbers
| Number | Prime factors |
|---|---|
| 12 | 2^2 × 3 |
| 18 | 2 × 3^2 |
| 24 | 2^3 × 3 |
| 30 | 2 × 3 × 5 |
| 36 | 2^2 × 3^2 |
| 42 | 2 × 3 × 7 |
| 48 | 2^4 × 3 |
| 50 | 2 × 5^2 |
| 60 | 2^2 × 3 × 5 |
| 64 | 2^6 |
| 72 | 2^3 × 3^2 |
| 84 | 2^2 × 3 × 7 |
| 90 | 2 × 3^2 × 5 |
| 96 | 2^5 × 3 |
| 100 | 2^2 × 5^2 |
| 120 | 2^3 × 3 × 5 |
| 144 | 2^4 × 3^2 |
| 180 | 2^2 × 3^2 × 5 |
| 200 | 2^3 × 5^2 |
| 210 | 2 × 3 × 5 × 7 |
| 256 | 2^8 |
| 360 | 2^3 × 3^2 × 5 |
| 500 | 2^2 × 5^3 |
| 720 | 2^4 × 3^2 × 5 |
| 1,000 | 2^3 × 5^3 |
| 1,024 | 2^10 |
| 2,026 | 2 × 1013 |
Frequently asked questions
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.
About this page
- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
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