Answer
\[
\boxed{2,\; 3,\; 5,\; 7,\; 11,\; 13,\; 17,\; 19,\; 23,\; 29,\; 31,\; 37,\; 41,\; 43,\; 47,\; 53,\; 59,\; 61,\; 67,\; 71,\; 73,\; 79,\; 83,\; 89,\; 97.}
\]
Work Step by Step
We can use the Sieve of Eratosthenes to identify all primes less than 100 by following these steps:
1. **List the Numbers:**
Write down all the integers from 2 to 99:
\[
2,\, 3,\, 4,\, 5,\, 6,\, 7,\, 8,\, 9,\, 10,\, 11,\, 12,\, \dots,\, 99.
\]
2. **Start with the First Prime (2):**
- Mark 2 as prime.
- Eliminate every multiple of 2 (other than 2 itself):
\(4,\, 6,\, 8,\, 10,\, 12,\, \dots,\, 98.\)
3. **Next Uncrossed Number (3):**
- The next number not eliminated is 3, so mark 3 as prime.
- Eliminate every multiple of 3 greater than 3:
\(6,\, 9,\, 12,\, 15,\, 18,\, \dots,\, 99.\)
(Some numbers like 6 and 12 may already have been removed.)
4. **Continue with the Next Uncrossed Number (5):**
- The next unmarked number is 5; mark it as prime.
- Eliminate its multiples:
\(10,\, 15,\, 20,\, 25,\, 30,\, \dots,\, 95.\)
5. **Next, 7:**
- The next unmarked number is 7; mark it as prime.
- Eliminate multiples of 7:
\(14,\, 21,\, 28,\, 35,\, 42,\, 49,\, 56,\, 63,\, 70,\, 77,\, 84,\, 91,\, 98.\)
6. **Numbers Greater Than 7:**
- Continue checking each number. Notice that once you reach numbers greater than \(\sqrt{100} \approx 10\), any composite number less than 100 will have already been marked as a multiple of a smaller prime.
- The numbers not eliminated in the list are the prime numbers.
7. **Final List of Primes Less Than 100:**
After completing the sieve, the numbers that remain unmarked are:
\[
2,\; 3,\; 5,\; 7,\; 11,\; 13,\; 17,\; 19,\; 23,\; 29,\; 31,\; 37,\; 41,\; 43,\; 47,\; 53,\; 59,\; 61,\; 67,\; 71,\; 73,\; 79,\; 83,\; 89,\; 97.
\]
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### Final Answer
The prime numbers less than 100 are:
\[
\boxed{2,\; 3,\; 5,\; 7,\; 11,\; 13,\; 17,\; 19,\; 23,\; 29,\; 31,\; 37,\; 41,\; 43,\; 47,\; 53,\; 59,\; 61,\; 67,\; 71,\; 73,\; 79,\; 83,\; 89,\; 97.}
\]
