Skip to main content
What are Prime Numbers?
- A prime number is a natural number greater than 1 that has only two factors: 1 and itself.
- In other words, it is divisible by exactly two distinct positive integers.
- Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17.
Properties of Prime Numbers
- 2 is the only even prime number.
- All other prime numbers are odd.
- There are infinitely many prime numbers.
- Every prime number greater than 3 can be expressed in the form 6n ± 1.
- 1 is not a prime number because it has only one factor.
Definition of Composite Numbers
- A composite number is a natural number greater than 1 that has more than two factors.
- Examples include 4, 6, 8, 9, 10.
- Every non-prime number (except 1) is a composite number.
Identification of Prime Numbers
- To identify whether a number is prime, check if it has any divisors other than 1 and itself.
- Use trial division method by dividing by prime numbers up to its square root.
- If no divisor is found, the number is prime.
Examples of Identification
- Check if 29 is prime: Divide by 2, 3, 5. No divisor found, so 29 is prime.
- Check if 35 is prime: Divisible by 5 (35 ÷ 5 =7), so 35 is composite.
Smallest Prime Numbers
- The smallest prime number is 2.
- The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Important Facts
- Every number greater than 1 is either prime or composite.
- 1 is neither prime nor composite.
- Prime numbers play a key role in cryptography, factorization, and number theory.
Properties of Composite Numbers
- Composite numbers have at least three factors.
- Smallest composite number is 4.
- All even numbers greater than 2 are composite.
Prime Number Tests
- For numbers up to 100, use Sieve of Eratosthenes to identify primes.
- For larger numbers, use trial division up to the square root.
- Advanced methods include Fermat’s test and Miller-Rabin test for very large primes.
Common Mistakes
- Thinking 1 is prime.
- Believing all odd numbers are prime (e.g., 9, 15 are composite).
- Forgetting that 2 is a prime number.
Miscellaneous Points
- Prime numbers are called the building blocks of natural numbers.
- The study of primes is part of arithmetic and algebra.
- Every composite number can be expressed as a product of prime numbers (prime factorization).