Prime numbers are the building blocks of all integers — every whole number greater than 1 is either prime itself or can be expressed as a product of primes. This property, called the Fundamental Theorem of Arithmetic, makes primes mathematically significant. Practically, they appear in cryptography, number theory and whenever you need to break a number down to its factors.
What Makes a Number Prime
A prime number has exactly two divisors: 1 and itself.
- Prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
- Not prime (composite): 4 (divisible by 2), 6 (by 2 and 3), 9 (by 3), 15 (by 3 and 5)...
- Neither prime nor composite: 1 (by definition)
The only even prime number is 2 — every other even number is divisible by 2.
How to Check if a Number is Prime
For small numbers (under 100): test divisibility by 2, 3, 5 and 7.
For larger numbers: only test divisibility by primes up to the square root of the number.
To check 97: √97 ≈ 9.8, so test primes up to 9: 2, 3, 5, 7.
- 97 ÷ 2 = 48.5 (not divisible)
- 97 ÷ 3 = 32.3 (not divisible)
- 97 ÷ 5 = 19.4 (not divisible)
- 97 ÷ 7 = 13.9 (not divisible)
- 97 is prime.
Prime Factorisation
Every composite number can be written as a product of primes:
- 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
- 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
This is useful for finding GCD and LCM — see how to find the LCM and GCD of numbers.
Why Primes Matter Beyond School
Modern internet encryption (including HTTPS on this site) relies on the difficulty of factoring very large numbers into their prime components. The security of RSA encryption, used in SSL certificates and secure messaging, depends on the fact that multiplying two large primes is easy, but factoring their product back into the original primes takes impractically long even for powerful computers.
For checking any specific number, use Prime Number Calculator.
Checking Primality for Larger Numbers
For numbers in the hundreds or thousands, mentally checking divisibility up to the square root becomes tedious. Prime Number Calculator confirms primality instantly and shows the prime factorisation of composite numbers. For finding GCD and LCM using prime factorisation — which provides an alternative method to the Euclidean algorithm — see how to find the LCM and GCD of numbers.
Twin Primes and Other Patterns
Twin primes are pairs of primes separated by 2: (3,5), (5,7), (11,13), (17,19), (29,31). Whether there are infinitely many twin primes is an unsolved problem in mathematics. Mersenne primes (of the form 2ⁿ − 1) are particularly large prime numbers sought by distributed computing projects. These patterns make primes a continuing area of mathematical research far beyond basic arithmetic.
