To find all composite numbers up to 19 using the Sieve of Eratosthenes, we start by listing all the numbers from 2 to 19. The process involves marking the multiples of each prime number, starting from the smallest prime (2).

1. List the numbers: $2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19$

2. Identify the primes:

   • Start with the first prime number, 2. Mark all multiples of 2 (greater than 2): $4, 6, 8, 10, 12, 14, 16, 18$ (these are composite).
   • Next, move to the next unmarked number, which is 3. Mark all multiples of 3 (greater than 3): $6, 9, 12, 15, 18$ (6, 12, and 18 are already marked).
   • Next, move to 5. Mark all multiples of 5 (greater than 5): $10, 15$ (10 is already marked).
   • Finally, the next number is 7. The only multiple is 14, which is already marked.

3. The numbers that remain unmarked (not composite) are the prime numbers.

The composite numbers up to 19 are: $4, 6, 8, 9, 10, 12, 14, 15, 16, 18$

Thus, the prime numbers up to 19 are: $2, 3, 5, 7, 11, 13, 17, 19$

Prime numbers up to 19: $2, 3, 5, 7, 11, 13, 17, 19$

Would you like any further details or have any questions? Here are some related questions:

1. What are the characteristics of prime numbers?
2. How can the Sieve of Eratosthenes be modified for larger numbers?
3. What are some applications of prime numbers in computer science?
4. How does the Sieve of Eratosthenes compare to other prime-finding algorithms?
5. Can you explain the concept of twin primes?

Tip: The Sieve of Eratosthenes is efficient for finding all primes up to a specified integer and can be implemented easily in programming languages.