Prime numbers

A prime number is an integer greater than 1 that has exactly two positive divisors: 1 and itself. The first several primes are 2, 3, 5, 7, 11, 13, 17, and 19. A whole number greater than 1 with more than two divisors — like 4, 6, 9, or 15 — is called composite. The number 1 is special: it is neither prime nor composite, because it has only one divisor.

Two facts about primes come up constantly. First, 2 is the only even prime — every other even number is divisible by 2. Second, the Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 factors into primes in exactly one way (apart from order): 60 = 2² × 3 × 5, and no other combination of primes multiplies to 60. This unique prime factorization is the engine behind finding greatest common factors, least common multiples, and counting a number's divisors.

To test whether a number is prime, you only need to check divisibility by primes up to its square root. For 97, checking 2, 3, 5, and 7 suffices, because any factor larger than √97 would pair with one smaller. Euclid proved more than two thousand years ago that the primes never run out — there are infinitely many.

Primes are a staple of the AMC competitions' number theory problems, the GRE quantitative section, and the Praxis Core math test. Expect questions on identifying primes, building prime factorizations, and using them to find GCFs, LCMs, and divisor counts — plus classic traps like forgetting that 1 is not prime and that 2 is prime.