Check if a number is prime and find its factors
Built & maintained by Pappu Venkata Subbi Reddy, founder of Clacify · Updated October 2026
Prime Number Checker tells you instantly whether any number is prime — and shows why, not just a yes/no answer. Enter a number and it explains the result, shows the full prime factorisation and every divisor if the number is composite, plus the five nearest primes below and above it, so you can see exactly where your number sits in the sequence. A prime number is a whole number greater than 1 that's divisible only by 1 and itself. It's built for students verifying homework, teachers preparing material, programmers implementing number-theory algorithms, and cryptography learners exploring why RSA encryption works. All computation — the primality test, the factorisation, and the nearest-prime search — runs locally in your browser, exactly (no rounding) for every whole number up to 9,007,199,254,740,991, with nothing sent to a server.
The checker uses trial division with two standard optimisations that make it dramatically faster than testing every possible divisor. First, it only tests divisors up to √n (the square root of the input) — if a number n has a divisor larger than its square root, it must also have a matching divisor smaller than the square root, so nothing above √n needs to be checked. Second, after handling 2 and 3 as special cases, every remaining prime is of the form 6k±1 (six times some integer, plus or minus one) — every other number is automatically divisible by 2 or 3 and can be skipped. Together these two optimisations mean checking a number up to millions completes in a fraction of a millisecond, entirely client-side. For numbers above one billion, primality is instead decided by a deterministic Miller–Rabin test using the first twelve primes (2 to 37) as witnesses — a combination proven to have no exceptions for any number below 3.3 × 10²⁴, so the answer is exact, not probabilistic, across the whole supported range. When a number is composite, trial division extracts its full prime factorisation by repeatedly dividing out the smallest prime factor until only 1 remains.
| Range | Primes | Count |
|---|---|---|
| 1–25 | 2, 3, 5, 7, 11, 13, 17, 19, 23 | 9 |
| 26–50 | 29, 31, 37, 41, 43, 47 | 6 |
| 51–75 | 53, 59, 61, 67, 71, 73 | 6 |
| 76–100 | 79, 83, 89, 97 | 4 |
There are exactly 25 prime numbers between 1 and 100, and exactly 168 prime numbers between 1 and 1,000 — both independently verified against the same algorithm this tool runs, not looked up. Enter any number above in the checker to see its own factorisation and neighbouring primes.
A prime number is a whole number greater than 1 that has exactly two positive divisors: 1 and itself. So 7 is prime (only 1 and 7 divide it evenly), but 8 is not (1, 2, 4, and 8 all divide it). Two specific numbers trip up almost everyone at some point. 1 is not prime — by definition a prime needs exactly two divisors, and 1 only has one (itself), so mathematicians classify it as neither prime nor composite, a category of its own. 2 is prime, and it is the only even prime — every other even number is divisible by 2, so it automatically fails the primality test. Numbers greater than 1 that aren't prime are called composite, and every composite number has a unique prime factorisation (this is the Fundamental Theorem of Arithmetic) — which is exactly what this tool computes and displays for you.
Take 221. Its square root is about 14.87, so under the √n rule we only need to test potential divisors up to 14 — anything larger can be ruled out immediately, since a factor pair can't both exceed the square root.
2 and 3 are ruled out first (221 is odd, and its digit sum 2+2+1=5 isn't divisible by 3). From there the 6k±1 pattern only requires testing 5, 7, 11, and 13. 221 ÷ 5 leaves a remainder, 221 ÷ 7 leaves a remainder, 221 ÷ 11 leaves a remainder — but 221 ÷ 13 = 17 exactly. The search stops there: 221 = 13 × 17, both prime, so 221 is composite with that exact factorisation.
Notice what just happened: instead of testing all 219 possible divisors from 2 to 220, the optimised algorithm only needed to test 4 numbers (5, 7, 11, 13) to find the answer. That gap — 4 tests instead of 219 — is why the square-root and 6k±1 optimisations matter, and why this tool returns an answer instantly even for numbers with far more digits than 221.
If a number n = a × b, then a and b can't both be greater than √n — because if they were, a × b would be greater than √n × √n = n, which is a contradiction. So for every factor pair of n, at least one factor is always ≤ √n. That guarantees: if no divisor up to √n evenly divides n, no divisor above √n can either (since it would need a matching partner below √n, which we already ruled out). This isn't an approximation or a shortcut with edge cases — it's a complete, exact test, which is why this tool's "prime" answer is always mathematically certain, never a probability.
Primes get rarer as numbers grow, but Euclid proved over 2,000 years ago — with a proof still taught essentially unchanged today — that there are infinitely many of them: assume a largest prime exists, multiply every prime up to it together and add 1, and the result must either be prime itself or have a prime factor larger than the assumed largest prime, a contradiction either way. Some primes cluster close together — "twin primes" like 11 and 13, or 17 and 19, differ by exactly 2 — while others have enormous gaps between them; whether infinitely many twin primes exist is still an open problem in mathematics (the Twin Prime Conjecture), unsolved despite centuries of attempts. Use the nearest-primes feature on this tool to explore gaps around any number yourself — some numbers have a twin prime one step away, others sit in a much sparser stretch.
Beyond number theory, primes are what makes RSA encryption — the algorithm behind HTTPS, online banking, and secure messaging — work at all. RSA relies on a simple asymmetry: multiplying two large primes together is fast (a computer does it instantly, as this tool demonstrates), but factoring that product back into its two original primes is, for sufficiently large primes, computationally infeasible even for the world's fastest supercomputers. A real RSA key uses primes hundreds of digits long specifically because the gap between "easy to multiply" and "practically impossible to factor" only becomes wide enough to be secure at that scale — the same trial-division principle this tool uses on small numbers would take longer than the age of the universe on a 300-digit number, which is precisely the point.
To check if n is prime, test divisibility by all prime numbers up to √n. If none divide n evenly, it is prime. Example: Is 97 prime? √97 ≈ 9.8. Check: 97 ÷ 2, 3, 5, 7 — none divide evenly → 97 is prime. Is 91 prime? 91 ÷ 7 = 13 → 91 = 7 × 13, not prime.
Prime factorisation expresses a number as a product of its prime factors. Every number has a unique prime factorisation (Fundamental Theorem of Arithmetic). Example: 360 = 2³ × 3² × 5. Used to find LCM/HCF, simplify fractions, and in cryptography (RSA encryption is based on the difficulty of factoring large numbers).