Consider the sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... Find the 100th prime number in this sequence.

To find the 100th prime number in the sequence of prime numbers, we can follow these steps:

1. ### Break Down the Problem

We need to identify the first 100 prime numbers and then select the 100th one from this list.

2. ### Relevant Concepts

• A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself.
• We can generate prime numbers using methods such as the Sieve of Eratosthenes or trial division.

3. ### Analysis and Detail

To find the first 100 prime numbers, we can list them out or utilize algorithms to generate them. Here’s a basic listing of the first few primes up to at least the 100th:

• The first few primes are:
  • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
  • 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
  • 73, 79, 83, 89, 97,
  • 101, 103, 107, 109, 113, 127, 131, 137, 139, 149,
  • 151, 157, 163, 167, 173, 179, 181, 191, 193, 197,
  • 199, 211, 223, 227, 229, 233, 239, 241, 251, 257,
  • 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
  • 317, 331, 337, 347, 349, 353, 359, 367, 373, 379,
  • 383, 389, 397, 401, 409, 419, 421, 431, 433, 439,
  • 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
  • 503, 509, 521, 523, 541

By counting these, we can see up to 541 is the 100th prime number.

4. ### Verify and Summarize

• The 100th prime number listed is 541.
• This can be verified through various prime number generating techniques or databases.

Final Answer

The 100th prime number is 541.