Why is prime factorisation unique?

The fundamental theorem of arithmetic guarantees that every integer above 1 has exactly one prime factorisation up to order. It is the reason the GCF and LCM can be read off the factorisations directly.

How do I count the divisors from the factorisation?

Add one to each exponent and multiply: for 360 = 2³ × 3² × 5 the count is (3+1)(2+1)(1+1) = 24. Each divisor chooses an exponent between zero and its maximum.

Why does factorisation matter for security?

Multiplying two large primes is easy, factoring the result back is very hard, and RSA encryption relies on that asymmetry. Records sit around 250 decimal digits for numbers with no special structure.

🔢 Prime Factorization Calculator

Find the prime factorization of a natural number.

Number to factorize

What is Prime Factorization?

A prime factor is a factor that is also a prime number. Prime factorization is the process of expressing a natural number as a product of prime numbers.

Factorization Method

Starting from the smallest prime (2), keep dividing until no further division is possible. Group repeated primes as powers.

360 = 2 × 180 = 2 × 2 × 90 = ...

360 = 2³ × 3² × 5

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be represented uniquely as a product of prime numbers (regardless of order).

n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ

Importance in Mathematics

It is the foundation for GCF/LCM calculation, high school and university mathematics, and the core principle of modern cryptography such as RSA.

Number of Divisors Formula

Number of divisors = (a₁+1)(a₂+1)…(aₖ+1)

e.g. 360 = 2³×3²×5¹ → number of divisors = (3+1)(2+1)(1+1) = 24

Example

360 = 2³ × 3² × 5, so it has 4 × 3 × 2 = 24 divisors.

97 is prime, so its only divisors are 1 and 97 and its factor tree has no branches.

Limits of This Calculator

Trial division is used, which is fast up to about 12 digits but slows down badly beyond; very large numbers need specialised factoring methods. Inputs must be positive integers of at least 2. The factorisation itself is unique, but the order in which primes are listed may differ from other tools.

Frequently Asked Questions

Why is prime factorisation unique?: The fundamental theorem of arithmetic guarantees that every integer above 1 has exactly one prime factorisation up to order. It is the reason the GCF and LCM can be read off the factorisations directly.
How do I count the divisors from the factorisation?: Add one to each exponent and multiply: for 360 = 2³ × 3² × 5 the count is (3+1)(2+1)(1+1) = 24. Each divisor chooses an exponent between zero and its maximum.
Why does factorisation matter for security?: Multiplying two large primes is easy, factoring the result back is very hard, and RSA encryption relies on that asymmetry. Records sit around 250 decimal digits for numbers with no special structure.