What is a composite number?

A **composite number** is a positive integer greater than 1 that has more than two distinct positive divisors. In other words, a composite number can be divided evenly by numbers other than 1 and itself. ### Understanding Composite Numbers - **Divisors:** The numbers that divide a given number exactly without leaving a remainder. For example, the divisors of 6 are 1, 2, 3, and 6. - **Prime Numbers vs. Composite Numbers:** - **Prime Number:** A positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. Example: 5 (divisors are 1 and 5). - **Composite Number:** As defined, it has more than two divisors. Example: 8 (divisors are 1, 2, 4, and 8). ### Examples of Composite Numbers - **4:** Divisors are 1, 2, and 4. - **9:** Divisors are 1, 3, and 9. - **12:** Divisors are 1, 2, 3, 4, 6, and 12. ### Key Points - **1 is Neither Prime nor Composite:** The number 1 has only one positive divisor (itself) and does not meet the criteria for either category. - **Prime Factorization:** Every composite number can be expressed as a product of prime numbers. For example, 12 can be factored into 2 × 2 × 3. - **Importance in Mathematics:** Understanding composite numbers is fundamental in number theory, especially in topics like prime factorization, least common multiples, and greatest common divisors. ### Identifying Composite Numbers To determine if a number is composite: 1. **Check for Divisors:** See if the number has any divisors other than 1 and itself. 2. **Use Factorization:** Break down the number into its prime factors. If there are multiple prime factors, the number is composite. **Example:** Is 15 a composite number? - **Divisors of 15:** 1, 3, 5, 15 - **Since it has more than two divisors**, 15 is a composite number. ### Summary Composite numbers play a crucial role in various areas of mathematics, especially in understanding the building blocks of numbers through prime factorization. Recognizing the difference between prime and composite numbers is essential for solving many mathematical problems and for advancing studies in number theory.