Prime Checker & Factorization

About this tool

The Prime Checker & Factorization tool determines whether an integer from 2 to 999,999,999 is prime and decomposes composite numbers into their prime factors using trial division. Type a number and press Check (or Enter). The result shows a PRIME or COMPOSITE badge; for composites it adds the factorization in prime-power form (for example 360 = 2³ × 3² × 5), the expanded factor list (2, 2, 2, 3, 3, 5) and every divisor in ascending order. For any input it also shows the previous prime and the next prime, which makes it handy for spotting twin primes and gaps. Everything runs in your browser, making this useful for both quick number-theory lookups and mathematics education.

Why use it

• Fast primality check and complete factorization in one tool.

• Lists every divisor of a composite number, not just its prime factors.

• Shows the previous and next prime for any input up to 999,999,999.

• Prime power notation and expanded factor list both shown simultaneously.

• Saves manually drawing factor trees on paper for homework verification.

• Eliminates calculator math errors that creep in during repeated division.

How to use

1. Enter an integer from 2 to 999,999,999 in the Number field.
2. Press Check or hit Enter.
3. Read the PRIME or COMPOSITE badge next to your number.
4. For composites, read the prime-power factorization, the expanded factor list and the full list of divisors.
5. Use the previous-prime and next-prime cards to find the nearest primes on either side.
6. Try entering small primes (2, 3, 5, 7) first to confirm the tool is working as expected before checking large numbers.

When it helps

• Factoring numbers in number theory coursework.
• Listing all divisors of a number for a homework or puzzle problem.
• Verifying prime factors before computing GCD or LCM.
• Checking whether a number is prime for mathematical curiosity.
• Testing whether a candidate number for a math puzzle is prime.
• Quickly determining the divisor count of a composite (multiply prime exponents + 1).

Examples

Tips

• Inputs are limited to 999,999,999 — for larger numbers use a big-integer tool or library.
• Draw a factor tree by hand from the factor list to teach why prime factorization is unique — every branch ends in primes, every time.
• When checking if a number is prime, you only need to test divisors up to its square root — the calculator does this automatically for speed.
• For GCD/LCM problems, factor both numbers first, then compare prime power exponents (minimum for GCD, maximum for LCM).
• Mersenne primes (2^p − 1) and Fermat primes (2^(2^n) + 1) are far beyond trial division — use a specialized library for cryptographic-scale primality testing.
• Twin primes (pairs differing by 2, like 11 and 13) are easy to spot using the previous- and next-prime cards.
• If a number ends in 0, 2, 4, 5, 6, or 8, it cannot be prime (except 2 and 5 themselves) — saves you from typing big composites.

Frequently Asked Questions

Glossary

Prime number

A natural number greater than 1 with exactly two distinct positive divisors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17.

Composite number

A natural number greater than 1 that has at least one divisor other than 1 and itself — i.e., it can be factored into smaller integers.

Trial division

The simplest primality test: try dividing n by every integer from 2 up to √n. If none divide evenly, n is prime.

Miller-Rabin test

A probabilistic primality test much faster than trial division for very large numbers; widely used in cryptographic libraries (RSA key generation).

Fundamental Theorem of Arithmetic

Every integer greater than 1 has a unique prime factorization (apart from order). This is what makes prime factorization meaningful.

Twin primes

A pair of primes that differ by exactly 2, such as (3,5), (11,13), or (17,19). Whether infinitely many twin primes exist is an unsolved problem.

Sieve of Eratosthenes

An ancient algorithm for finding all primes up to a fixed bound by iteratively eliminating multiples — efficient for generating prime tables.

Coprime (relatively prime)

Two integers are coprime if their greatest common divisor is 1 — i.e., they share no prime factors. Useful in modular arithmetic and RSA.