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 1, itself, and at least one other positive integer. This distinguishes composite numbers from **prime numbers**, which have exactly two distinct positive divisors: 1 and themselves. ### Key Points: - **Definition**: A composite number has at least one divisor other than 1 and itself. - **Examples**: - **4**: Divisors are 1, 2, and 4. - **6**: Divisors are 1, 2, 3, and 6. - **9**: Divisors are 1, 3, and 9. - **Non-Examples**: - **2, 3, 5, 7, 11, ...**: These are prime numbers because they have no divisors other than 1 and themselves. - **1**: By definition, 1 is neither prime nor composite. ### Visual Representation: Consider the number **8**: - **Divisors of 8**: 1, 2, 4, 8 - Since 8 has more than two divisors, it is a composite number. ### Importance in Mathematics: Composite numbers are fundamental in number theory, especially in topics like factorization, where understanding the building blocks of numbers (prime factors) relies on distinguishing between prime and composite numbers. ### Prime Factorization: Every composite number can be expressed as a product of prime numbers. This is known as its prime factorization. For example: - **12** can be factored into \(2 \times 2 \times 3\) or \(2^2 \times 3\). - **45** can be factored into \(3 \times 3 \times 5\) or \(3^2 \times 5\). Understanding composite numbers and their properties is essential for various applications in mathematics, including simplifying fractions, finding least common multiples, and solving equations.