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Prime factorization is one of the most fundamental concepts in number theory — the mathematical study of integers and their properties. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a unique product of prime numbers. This uniqueness makes prime factorization indispensable across mathematics and computer science.

This calculator uses trial division — testing divisibility by each prime starting from 2 — to systematically find all prime factors. The result is displayed in exponential form and as a complete step-by-step division walkthrough.

In the digital age, prime factorization has taken on critical importance in cryptography. The RSA algorithm — which secures most internet communications — relies on the mathematical fact that while multiplying two large primes is trivial, factoring their product is computationally infeasible for large enough numbers.

공식

Every integer n > 1 can be uniquely written as:

n = p₁^a₁ × p₂^a₂ × ··· × pₖ^aₖ

where p₁ < p₂ < ··· < pₖ are distinct primes and a₁, a₂, ..., aₖ ≥ 1.

Number of divisors: τ(n) = (a₁+1)(a₂+1)···(aₖ+1)

계산 방법

Enter any positive integer from 2 to 10,000,000.
The algorithm starts by dividing by 2, the smallest prime.
Each time the number divides evenly, the divisor is recorded as a factor.
When 2 no longer divides evenly, the algorithm tries 3, 5, 7, 11, ...
This continues until the remaining number equals 1 or is itself prime.
The factorization is written in exponential form: n = p₁^a₁ × p₂^a₂ × ...
Divisors are counted using the formula τ(n) = (a₁+1)(a₂+1)···(aₖ+1).

예시

Factorize 360: 360÷2=180 → 180÷2=90 → 90÷2=45 → 45÷3=15 → 15÷3=5 → 5 is prime. Result: 360 = 2³ × 3² × 5¹. Divisors: (3+1)(2+1)(1+1) = 24.

주요 용어 설명

Prime number: An integer > 1 with no factors other than 1 and itself
Composite number: An integer > 1 that is not prime
Fundamental Theorem of Arithmetic: Every integer > 1 has a unique prime factorization
Trial division: Factorization algorithm testing divisibility by successive primes
Exponent: In p^a, the exponent a counts how many times prime p appears
Divisor function τ(n): Counts the total number of positive divisors of n

일반적인 사용 사례

Simplifying fractions to lowest terms
Finding GCD and LCM of numbers
RSA encryption and public-key cryptography
Solving number theory problems in mathematics competitions
Understanding divisibility rules in mathematics
Algebraic factoring and polynomial simplification

자주 묻는 질문

What is prime factorization?

Prime factorization expresses a number as a product of its prime factors. Every integer > 1 has a unique prime factorization — the Fundamental Theorem of Arithmetic. Example: 360 = 2³ × 3² × 5.

What is a prime number?

A prime number is a natural number greater than 1 with no positive divisors other than 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23... There are infinitely many primes (Euclid, ~300 BC).

How is prime factorization used in real life?

Prime factorization is used in RSA cryptography (internet security), simplifying fractions, finding GCD and LCM, and number theory. Modern internet security relies on the difficulty of factoring large numbers.

How do I count divisors from prime factorization?

If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, then the number of positive divisors is (a₁+1)(a₂+1)···(aₖ+1). Example: 12 = 2² × 3¹ has (2+1)(1+1) = 6 divisors.

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