Free prime number calculator: check if a number is prime, find prime factors, and list prime numbers in a range. Essential for number theory and cryptography.
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A prime p has exactly two divisors: 1 and p itselfpThe number being tested√pOnly need to test divisors up to √pTo test primality, we check if any integer from 2 to √n divides n evenly. If none do, n is prime. This works because if n has a factor larger than √n, it must also have one smaller than √n. For prime factorization, we repeatedly divide by the smallest prime factor.
Prime numbers are the building blocks of all integers - every whole number greater than 1 can be expressed as a unique product of primes. This fundamental property, known as the Fundamental Theorem of Arithmetic, makes primes central to number theory, cryptography, and computer science. Whether you are a student studying divisibility or a developer working with cryptographic algorithms, understanding prime numbers is essential.
Enter any positive integer to instantly determine whether it is prime or composite. The calculator also provides the complete prime factorization, lists all divisors, and counts the total number of divisors. The step-by-step explanation shows the trial division method used to test primality.
Testing 97 for primality using trial division.
97 is prime — it has exactly two divisors.
Breaking 84 into its prime building blocks.
84 = 2 × 2 × 3 × 7 — composite, with 12 total divisors.
Always divide out the smallest prime factor first, then repeat on what's left.
This tool is invaluable for students learning number theory, programmers implementing cryptographic functions, and puzzle enthusiasts working with mathematical problems. It handles numbers up to several million efficiently using the optimized trial division algorithm.
Remember that 1 is neither prime nor composite - it is a special case. Also, 2 is the only even prime; all other even numbers are composite. When performing prime factorization manually, always start with the smallest prime factor (2) and work upward.
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