Prime Number Checker
Check if a number is prime, find all prime factors, and list all primes within a range. Instant results with factorization breakdown.
Is this number prime?
How to Use This Tool
- Choose a mode: "Is it Prime?" to check a single number, "Prime Factorization" to break a number into its prime factors, or "Primes in Range" to list all primes between two numbers.
- Enter the number (or range) into the input field.
- The result updates instantly as you type.
- For a non-prime number, review the prime factorization and the list of all divisors shown below the result.
Formula & How It Works
Trial Division Primality Test
n is prime if no integer i in [2, √n] divides n evenlyTest n for divisibility by every integer from 2 up to the square root of n; if none divide evenly, n is prime. Testing beyond √n is unnecessary because any factor larger than √n must pair with a factor smaller than √n.
Prime Factorization
n = p₁^e₁ × p₂^e₂ × … × pₖ^eₖEvery integer greater than 1 can be written as a unique product of prime numbers raised to whole-number powers.
Divisor Pairing
If i divides n, then n / i also divides nDivisors are found by testing integers up to √n; each divisor found below the square root pairs with a corresponding divisor above it.
Practical Examples & Common Use Cases
Example: Is 97 Prime?
Testing divisors from 2 up to √97 ≈ 9.8 (i.e. 2, 3, 5, 7): none divide 97 evenly, so 97 is prime.
Example: Prime Factorization of 360
360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Example: Primes Between 1 and 20
Checking each number from 1 to 20 gives the primes: 2, 3, 5, 7, 11, 13, 17, 19.
Frequently Asked Questions
A prime number is greater than 1 and has exactly two distinct divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
No. By mathematical convention, 1 is not considered prime because prime numbers are defined as having exactly two distinct divisors, and 1 only has one divisor (itself).
Prime factorization breaks a number into a product of prime numbers. Every integer greater than 1 has a unique prime factorization. Example: 60 = 2² × 3 × 5.
To check if n is prime, test divisibility by all integers up to √n. If none divide evenly, the number is prime. This calculator uses that method.
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