What it is
A prime number is a whole number greater than 1 with exactly two positive divisors: 1 and the number itself. 7 is prime, because the only numbers that divide 7 evenly are 1 and 7. 12 is not, because 12 also divides by 2, 3, 4 and 6.
A whole number greater than 1 that is not prime is composite: it can be written as a product of two smaller whole numbers greater than 1, as 12 = 3 · 4. The numbers 0 and 1 are neither prime nor composite.
The first primes are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, …
There are 25 primes below 100 and 168 below 1,000. All of them are in the table of primes.
- 2 is prime, and the only even prime. Every other even number divides by 2.
- 1 is not prime. It has one positive divisor, not two.
- 51 is not prime: 51 = 3 · 17. Its digit sum 5 + 1 = 6 divides by 3, so the number does too — see the divisibility rules.
- 97 is prime. It is enough to try the primes up to the square root of 97, which are 2, 3, 5 and 7, and none of them divides it.
Trying up to the square root is enough because divisors come in pairs. In 91 = 7 · 13 one factor is at most √91 ≈ 9.5, and the same holds for every way of splitting a number into two factors. If there is no divisor below the root, there is none above it either.
Why it matters
Primes are the building blocks of the whole numbers. Every whole number greater than 1 is either a prime or a product of primes, and the product is the same however you arrive at it, apart from the order: 360 is 2³ · 3² · 5 and nothing else. That is the fundamental theorem of arithmetic, and prime factorisation is the method that finds the product.
Most of what you do with divisibility goes through those factors:
- The greatest common divisor is the primes two numbers share — see GCD.
- The least common multiple takes every prime found in either number — see LCM. It gives the lowest common denominator when fractions are added.
- Simplifying a fraction is striking out primes found in both numerator and denominator: 84/360 = (2² · 3 · 7)/(2³ · 3² · 5) = 7/30.
Outside school, a widely used kind of encryption, RSA, rests on the fact that multiplying two large primes is fast while recovering them from the product is, with the methods known, very slow.
The prime factorisation calculator splits any number and shows each step, and the worksheets give questions with an answer key.
Common misconceptions
- “1 is prime.” It has only one divisor. If it counted as prime, factorisation would stop being unique: 6 = 2 · 3 = 1 · 2 · 3 = 1 · 1 · 2 · 3.
- “Odd numbers are prime.” 9, 15, 21, 25 and 27 are odd and composite. A prime above 2 has to be odd, but far from every odd number is prime.
- “A number that looks unusual is probably prime.” 91 = 7 · 13 and 57 = 3 · 19 look prime and are not. Try the divisions rather than guess.
- “The primes run out eventually.” They thin out the further you go, but they never run out. Euclid showed it more than 2,000 years ago.
Common questions
How many primes are there?
Infinitely many. Euclid’s argument: multiply some primes together and add 1. The new number does not
divide by any of them, since there is always 1 left over, so its prime factors are primes that were
not on the list. The number itself need not be prime: 2 · 3 · 5 · 7 · 11 · 13 + 1 = 30,031, which is
59 · 509.
Are negative numbers prime?
No, not with the definition here. A prime is a whole number greater than 1, so neither negative
numbers, 0 nor 1 count.
How do you tell quickly whether a number is prime?
Divide by the primes 2, 3, 5, 7, 11, … as long as the prime squared is not larger than the number. If
none divides it evenly, the number is prime. For 2, 3 and 5 it is enough to look at the digits — see
the divisibility rules.