UtilityKit

500+ fast, free tools. Most run in your browser only; Image & PDF tools upload files to the backend when you run them.

Prime Checker & Factorization

Check whether a number is prime and find its prime factorization with a visual factor tree.

About Prime Checker & Factorization

The Prime Checker & Factorization tool determines whether any integer is prime and decomposes composite numbers into their prime factors using trial division. Enter any positive integer and the tool reports whether it is prime (with a brief explanation of why it passed primality tests) or composite (with its complete prime factorization in canonical form, e.g. 360 = 2³ × 3² × 5). The factorization is shown both in prime power notation and as an expanded factor list. A visual factor tree diagram illustrates the recursive decomposition step by step, making this useful for both quick number-theory lookups and mathematics education.

Why use Prime Checker & Factorization

  • Fast primality check and complete factorization in one tool.
  • Visual factor tree is great for teaching and learning number theory.
  • Works for large numbers well into the billions using trial division.
  • Prime power notation and expanded 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 Prime Checker & Factorization

  1. Enter any positive integer in the input field.
  2. The primality result and factorization appear instantly.
  3. Toggle 'Factor Tree' to see the visual decomposition diagram.
  4. Use the Copy button to copy the factorization in prime power notation.
  5. For large numbers, allow a few seconds for trial division to complete before retrying.
  6. Use the result alongside a GCD calculator — paste both factorizations and compare overlapping prime powers.
  7. Try entering small primes (2, 3, 5, 7) first to confirm the tool is working as expected before checking large numbers.

When to use Prime Checker & Factorization

  • Factoring numbers in number theory or cryptography coursework.
  • Teaching prime factorization with the visual factor tree.
  • Verifying prime factors before computing GCD or LCM.
  • Checking whether a large 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

Small composite

Input: 360

Output: 360 = 2³ × 3² × 5 (composite, 24 divisors)

Prime number

Input: 97

Output: 97 is prime (no divisors other than 1 and 97)

Large semiprime

Input: 2147483647

Output: 2147483647 is prime (Mersenne prime M31 = 2³¹ − 1)

Power of two

Input: 1024

Output: 1024 = 2¹⁰ (composite)

Tips

  • For very large numbers near JavaScript's safe integer limit (2^53), trial division can take noticeable time — start with smaller numbers to confirm the format before pushing to billions.
  • Use the factor tree to teach kids and students 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 this tool — try sequential entries to find them.
  • 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

What makes a number prime?▾
A prime number has exactly two distinct positive divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13...
Is 1 a prime number?▾
No. 1 is neither prime nor composite by definition. Primes must have exactly two distinct divisors; 1 has only one.
What is prime factorization?▾
Prime factorization expresses a composite number as a product of prime numbers. For example, 60 = 2² × 3 × 5. The Fundamental Theorem of Arithmetic guarantees this is unique for every positive integer.
What is the largest number this can handle?▾
The tool handles numbers up to JavaScript's Number.MAX_SAFE_INTEGER (2^53 - 1 ≈ 9 quadrillion). For very large numbers close to this limit, factorization may be slow.
What is a factor tree?▾
A factor tree is a diagram that shows how a number breaks into factors, then each factor breaks further until all branches end at prime numbers.
How does this tool compare to Wolfram Alpha for primality testing?▾
For numbers up to ~10^15, results match Wolfram Alpha exactly using trial division. Wolfram Alpha uses Miller-Rabin and other probabilistic tests for cryptographic-scale primes (100+ digits) which this tool does not target.
Are my input numbers logged or stored anywhere?▾
No. The factorization runs entirely in your browser using JavaScript — the input never leaves your device. Safe for use with sensitive data like cryptographic seed numbers.
What is the time complexity of the algorithm?▾
Trial division is O(√n) per check. For n = 10^12 that means up to 10^6 divisions — completes in milliseconds. For n approaching 2^53, expect a few seconds in worst case.

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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.