Prime numbers are fundamental building blocks in mathematics. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. For example, 7 is prime because its only divisors are 1 and 7. In contrast, 6 is composite because it has divisors 1, 2, 3, and 6. Remember that 1 is neither prime nor composite, and 2 is the only even prime number.
Now let's systematically identify the first fifteen prime numbers. We start with 2, which is the only even prime number. Then we have 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Notice that except for 2, all primes are odd numbers. For each number, we test divisibility by smaller numbers. For example, to verify that 11 is prime, we check if it's divisible by 2, 3, or 5, and find that none divide evenly, confirming 11 has only two divisors: 1 and itself.
The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a given limit. We start by listing all numbers from 1 to 100. First, we cross out 1 since it's neither prime nor composite. Then we highlight 2 as prime and cross out all its multiples in red. Next, we move to 3, mark it as prime, and cross out its remaining multiples in blue. We continue with 5, crossing out its multiples in green, then 7 in orange. After processing all numbers up to the square root of 100, the remaining unmarked numbers are all prime.
When testing larger numbers for primality, we use efficient methods to avoid checking all possible divisors. The key insight is the square root rule: to test if n is prime, we only need to check divisors up to the square root of n. This works because if n equals a times b, and a is greater than the square root of n, then b must be less than the square root, so we would have already found b as a divisor. Let's test 97: its square root is approximately 9.8, so we check divisors 2, 3, 5, 7, and 9. None of these divide 97 evenly, so 97 is prime. In contrast, when testing 91, we find that 91 divided by 7 equals 13, so 91 is composite.