Prime Number Checker

Quick start

1. Type a whole number. The answer updates as you type.
2. For a composite number, read the prime factorization and every divisor.
3. Check the previous and next prime to see where your number sits.

Prime, composite, or neither

A prime has exactly two divisors: 1 and itself. 13 and 101 are prime. A composite number has more than two divisors, such as 15 = 3 × 5. The number 1 has only one divisor, so it is neither. 2 is the only even prime.

Why testing up to the square root is enough

If n splits as a × b, the two factors cannot both be bigger than √n, because then their product would exceed n. So if nothing up to √n divides n, nothing larger will either. To test 221, you only need primes up to 14. Trying 13 gives 221 = 13 × 17, so it is composite.

Counting divisors from the factorization

Write the number as primes with powers, add 1 to each power and multiply. 180 = 2² × 3² × 5, so it has (2 + 1)(2 + 1)(1 + 1) = 18 divisors.

Primes up to 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. That is 15 primes.

Common questions