How to Check if a Number Is Prime (Fast Methods, 2026)

To check if a number is prime, test whether any whole number from 2 up to its square root divides it evenly. If none do, the number is prime; if one does, it is composite. That single shortcut is the heart of every fast primality test you will use by hand.

This guide shows the square-root method, the divisibility shortcuts that let you reject most numbers in seconds, and how to read the result — with worked examples you can reproduce in the Prime Number Checker.

What makes a number prime?

A prime number is a whole number greater than 1 that has exactly two divisors: 1 and itself. 2, 3, 5, 7, 11 and 13 are prime. A composite number has at least one extra divisor — 12 is composite because 2, 3, 4 and 6 all divide it. The numbers 0 and 1 are neither prime nor composite.

The square-root rule

You never need to test divisors beyond √n. If a number n had a factor larger than its square root, that factor would have to pair with one smaller than the square root — which you would already have found. So to check 97, you only test 2, 3, 5 and 7 (since 7² = 49 and 11² = 121 > 97). None divide 97, so 97 is prime.

Divisibility shortcuts

Most non-primes fail one of these instantly:

• Even? Divisible by 2 (except 2 itself).
• Digit sum divisible by 3? Then the number is too.
• Ends in 0 or 5? Divisible by 5.

Only after these do you test 7, 11, 13 and so on up to the square root.

Worked examples

Is 91 prime? It is odd, digit sum 10 (not divisible by 3), does not end in 0/5 — but 7 × 13 = 91, so it is composite. Is 101 prime? Test 2, 3, 5, 7 (7² = 49, next is 11² = 121 > 101): none divide it, so 101 is prime. The Prime Number Checker shows the smallest divisor and the nearest primes for any number.

Common mistakes

The big ones: calling 1 prime (it is not), forgetting that 2 is the only even prime, and testing divisors all the way to n instead of stopping at √n. For deeper study, see the Sieve of Eratosthenes.