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).

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