# The Prime Page

First, the basic definition:

A prime number is a natural number with exactly two positive divisors: itself and 1.

Some examples are 2, 7, 97, and 2729. (Note that with only one divisor, 1 is not a prime.)

Prime numbers are fascinating on their own, but have lots of cool and scary applications!

## How do you tell "primes" from "non-primes"?

If I give you a small number, like 15, you can tell it's not prime because 15 = 3 × 5, so that 15 has divisors other than 1 and 15 itself. Numbers like this are called composite numbers.

If you have a larger number like 91 or 97 you have to work a little; try dividing in small primes in order:

2, 3, 5, 7, 11, . . . (see table below).

If any of them go into your number evenly, it's composite.

With 91 you try 2, 3, and 5; they don't go in evenly (they leave a remainder); but 91 ÷ 7 = 13.

This means 91 = 7 × 13, so it's not a prime after all.

What about 97? None of those little primes 2, 3, 5, or 7 go into 97. When can we stop? At 97? 50?