Know the definition first
A prime number is a whole number greater than 1 with exactly two positive divisors: 1 and the number itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is neither prime nor composite, and 2 is the only even prime.
Check only up to the square root
To test a positive integer n, you do not need to try every number below n. If n has a factor larger than its square root, it must also have a matching factor smaller than the square root. So testing possible divisors up to the square root is enough.
- If the number is less than 2, it is not prime.
- Check whether it is divisible by 2. If it is even and greater than 2, it is not prime.
- Try odd divisors from 3 up to the square root. If any divides evenly, the number is composite.
- If none divides evenly, the number is prime.
Example: is 97 prime?
The square root of 97 is a little less than 10. Check the possible prime divisors up to that point: 2, 3, 5 and 7. The number is not divisible by any of them, so 97 is prime.
For 91, the check finds 7: 91 ÷ 7 = 13. That is enough to show that 91 is composite; there is no need to test larger divisors.
Common mistakes
- Calling 1 prime. A prime must have exactly two positive divisors, while 1 has only one.
- Assuming every odd number is prime. Odd composites such as 9, 15, 21 and 91 have odd divisors.
- Testing only a few small divisors for a large number. The check must cover every possible divisor up to the square root, or use a verified primality algorithm.
UsefulVerse checks practical whole-number inputs locally in your browser. A prime result is a divisibility check, not a proof about a cryptographic key or a substitute for specialized number-theory software.