素因数分解計算機

任意の数の素因数分解を段階的に表示します。素因数、指数、約数の個数を計算。

素因数分解計算機とは？

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.

計算式

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)

計算方法

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).

例

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.

重要な用語の説明

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

よくある使用例

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

よくある質問

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.

素因数分解計算機