🔢 Prime Number Checker
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📌 Understanding Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
- Basic Trial Division: Checks every number from 2 to n–1.
- Optimized Trial Division: Checks divisibility only up to √n, reducing unnecessary checks.
- Sieve Method: Generates all prime numbers up to a limit using the Sieve of Eratosthenes, then checks if the number is in that list.
📌 Examples of Prime Numbers
| Number | Is Prime? | Method |
|---|---|---|
| 2 | Yes | Any |
| 15 | No | Any |
| 29 | Yes | Any |
| 100 | No | Any |
🔧 Practical Applications of Prime Numbers
1. Cryptography: Prime numbers are essential in encryption algorithms.
2. Mathematics: They play a key role in number theory and various proofs.
3. Computer Science: Used in hashing, randomized algorithms, and more.
❓ FAQs
Q1: What is a prime number?
✅ A prime number is a natural number greater than 1 with exactly two distinct positive divisors: 1 and itself.
Q2: Which method is the fastest?
✅ For checking a single number, the optimized trial division (up to √n) is very efficient.
Q3: When should I use the Sieve method?
✅ The sieve method is ideal if you need to generate all primes up to a certain limit (here n ≤ 1000). It’s less efficient for a single, very large number.
Q4: Can this checker handle non-integer inputs?
✅ No, please enter a positive integer greater than 1.
Q5: Why is checking up to √n sufficient?
✅ If n has a factor larger than √n, it must also have a factor smaller than √n, so checking up to √n covers all possibilities.