Prime Factorization Calculator
Find the prime factorization of any positive integer. Break down numbers into their prime factors and visualize the factor tree with step-by-step decomposition.
Number Input
Prime Factorization
Factor Tree
Step-by-Step Factorization
Methods to Find Prime Factorization
1. Trial Division Method
Start with the smallest prime number (2) and divide the number repeatedly until it's no longer divisible. Then move to the next prime (3, 5, 7, 11, ...) and repeat the process.
Result: 60 = 2² × 3 × 5
2. Factor Tree Method
Break the number into any two factors, then continue breaking down composite factors until only prime factors remain. This creates a tree-like structure showing the factorization process.
3. Prime Factorization Algorithm
An optimized approach that only tests divisibility by prime numbers up to √n, significantly reducing the number of operations needed for large numbers.
How to Use This Calculator
Understanding Prime Numbers and Factorization
What are Prime Numbers?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Fundamental Theorem of Arithmetic
Every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This is the foundation of prime factorization.
12 = 2² × 3
30 = 2 × 3 × 5
100 = 2² × 5²
Types of Numbers
Prime Numbers
Numbers with exactly two factors: 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Composite Numbers
Numbers with more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
Perfect Powers
Numbers that can be expressed as a^n where a and n are positive integers. Examples: 4 = 2², 8 = 2³, 9 = 3².
Highly Composite
Numbers with more divisors than any smaller positive integer. Examples: 1, 2, 4, 6, 12, 24, 36, 48.
Example Calculations
Example 1: Small Composite Number
Example 2: Power of Prime
Example 3: Multiple Prime Factors
Frequently Asked Questions
What is prime factorization?
Prime factorization is the process of breaking down a composite number into a product of prime numbers. It's like finding the "building blocks" of a number using only prime numbers.
Why is prime factorization important?
Prime factorization is fundamental in number theory, cryptography, finding GCD and LCM, simplifying fractions, and solving various mathematical problems. It's also crucial in computer science and security.
What if I enter a prime number?
If you enter a prime number, the calculator will show that the number itself is its only prime factor. Prime numbers cannot be factored further since they have no divisors other than 1 and themselves.
How large numbers can this calculator handle?
The calculator can handle numbers up to several million efficiently. For very large numbers (billions or more), the calculation may take longer due to the computational complexity of finding large prime factors.