Prime Factorization Calculator
Find the prime factors and prime power representation of any number
Prime Factorization
(Prime Number)
List of Prime Factors
Step-by-Step Solution
Prime vs Composite
A **prime number** is a whole number greater than 1 whose only factors are 1 and itself. Examples: 2, 3, 5, 7, 11...
A **composite number** is a whole number greater than 1 that has more than two factors. Every composite number can be uniquely expressed as a product of primes.
Trial Division Method
The simplest way to find prime factors is the **trial division** method:
- • Divide the number by the smallest prime (2).
- • If it divides evenly, the prime is a factor.
- • Repeat with the quotient until no longer divisible.
- • Move to the next prime (3, 5, 7...) and repeat.
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Detailed Information
What is Prime Factorization?
Prime factorization is the unique way of expressing any integer greater than 1 as a product of prime numbers. It is a fundamental concept in number theory used in cryptography, computer science, and mathematics education. According to the Fundamental Theorem of Arithmetic, every positive integer has exactly one prime factorization (ignoring the order of the factors).
How to Use This Calculator
- Enter a Number: Type any positive integer (greater than 1) into the input field.
- View Results: The calculator will instantly display the prime factors, the prime power representation, and a step-by-step breakdown of the division process.
- Reset: Use the 'Calculate Again' button to clear the input and start over.
Understanding the Results
- Prime Factors: The list of prime numbers that, when multiplied together, equal the original number. For example, the prime factors of 12 are 2, 2, and 3.
- Prime Power Representation: A concise way to write the factorization using exponents. For 12, this is 2² × 3.
- Prime Number: If a number is only divisible by 1 and itself, it will show as its own only prime factor.
Formula and Method
We use the Trial Division method:
- Start with the smallest prime, 2.
- Divide the number by the prime as many times as possible until it no longer divides evenly.
- Move to the next prime (3, 5, 7...) and repeat the process.
- The process stops when the remaining number is also prime.
Frequently Asked Questions (FAQ)
Is 1 a prime factor?
No, 1 is neither prime nor composite, so it is not included in prime factorization.
Can negative numbers be factored into primes?
Technically, prime factorization is defined for positive integers. However, for a negative number, you can factor it as -1 multiplied by the prime factors of its absolute value.
What are coprime numbers?
Two numbers are coprime if their only common prime factor is none, meaning their Greatest Common Divisor (GCD) is 1.
[!NOTE] Disclaimer: This calculator is for educational purposes. While we strive for absolute accuracy, please verify important results for critical mathematical or cryptographic applications.