Definition of Composite Numbers

  • A composite number is a natural number greater than 1 that has more than two factors.
  • This means it can be divided by 1, itself, and at least one other number.
  • Examples include 4, 6, 8, 9, 10, 12, 15.

Basic Properties

  • The smallest composite number is 4.
  • All natural numbers greater than 1 are either prime or composite.
  • 1 is neither prime nor composite.
  • Every composite number has at least three factors.

Even and Odd Composite Numbers

  • All even numbers greater than 2 are composite.
  • Example: 4, 6, 8, 10.
  • Odd composite numbers are those odd numbers with more than two factors.
  • Example: 9 (3 × 3), 15 (3 × 5).

Factors and Divisibility

  • A composite number can be factored into smaller numbers other than 1 and itself.
  • Every composite number can be expressed as a product of prime numbers (prime factorization).
  • Example: 12 = 2 × 2 × 3.
  • Composite numbers have at least one divisor other than 1 and itself.

Prime Factorization Property

  • The Fundamental Theorem of Arithmetic states every composite number has a unique prime factorization (except for the order of factors).
  • Example: 18 = 2 × 3 × 3.

Identification of Composite Numbers

  • If a number has more than two factors, it is composite.
  • If a number can be divided evenly by any integer other than 1 and itself, it is composite.
  • Example: 21 is divisible by 3 and 7.

Special Cases

  • 1 is neither prime nor composite.
  • 2 is the only even prime number, and not composite.
  • All other even numbers >2 are composite.

Properties Related to Multiples

  • Every composite number is a multiple of a smaller natural number other than 1.
  • Example: 20 is a multiple of 4 and 5.
  • Composite numbers are useful in finding LCM and HCF.

Square and Cube Composites

  • The square of any prime number is always composite.
  • Example: 5² = 25 (composite).
  • Any number with a square root that is a whole number (except primes) is composite.

Composite Number Examples

  • 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
  • All have more than two factors.

Common Mistakes to Avoid

  • Thinking 1 is composite (it is not).
  • Forgetting that 2 is prime, not composite.
  • Assuming all odd numbers are prime (e.g., 9 is composite).

Miscellaneous Facts

  • The study of composite numbers is a part of number theory.
  • Composite numbers are essential in mathematics, coding, and cryptography.
  • Learning to quickly identify composite numbers improves speed and accuracy in exams.

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