Composite Number || Even & Odd Number || Coprime Number || Twin Prime Number || 6th Class

A composite number is a natural number greater than 1 that is not a prime number, meaning it has more than two positive divisors. In other words, a composite number can be divided evenly by at least one positive integer other than 1 and itself. For example, 4 is a composite number because it can be divided by 1, 2, and 4. Similarly, 6 is composite because it has divisors 1, 2, 3, and 6. Unlike prime numbers, which have only two divisors (1 and the number itself), composite numbers have multiple factors. Composite numbers can always be factored into smaller integers that are prime. For instance, 12 can be factored into 2 × 2 × 3, where 2 and 3 are prime numbers. This property is essential in number theory and is a key aspect of the Fundamental Theorem of Arithmetic, which states that every positive integer greater than 1 is either a prime number or can be factored uniquely as a product of prime numbers (ignoring the order of the factors). Composite numbers are contrasted with prime numbers and play a crucial role in various mathematical concepts, including factorization, divisibility, and algebra. Even and odd numbers are classifications of integers based on their divisibility by 2. **Even Numbers**: These are integers that can be exactly divided by 2, leaving no remainder. Mathematically, an even number can be expressed as \( 2n \), where \( n \) is any integer. Examples include 0, 2, 4, 6, 8, 10, etc. **Odd Numbers**: These are integers that cannot be exactly divided by 2. They leave a remainder of 1 when divided by 2. An odd number can be expressed as \( 2n + 1 \), where \( n \) is any integer. Examples include 1, 3, 5, 7, 9, 11, etc. *Coprime numbers* (also known as *relatively prime* or **mutually prime numbers**) are two or more integers that have no common factors other than 1. In other words, their greatest common divisor (GCD) is 1. Key Points: **Coprime pairs**: For example, the numbers 8 and 15 are coprime because the only factor they share is 1. **Multiple numbers**: More than two numbers can also be coprime. For instance, 6, 35, and 13 are coprime as a group because they share no common factor other than 1. **Prime numbers**: Any two distinct prime numbers are always coprime. For example, 3 and 7 are coprime. Coprime numbers do not need to be prime individually. For instance, 14 and 15 are coprime, even though neither is a prime number. *Twin primes* are pairs of prime numbers that have a difference of exactly 2. In other words, if \( p \) is a prime number, and \( p + 2 \) is also a prime number, then \( p \) and \( p + 2 \) form a twin prime pair. Examples: The pair (3, 5) is a twin prime because both 3 and 5 are prime, and their difference is 2. The pair (11, 13) is another twin prime pair. The pair (17, 19) is also a twin prime pair. Twin primes are of interest in number theory because, despite extensive research, it remains an open question whether there are infinitely many twin primes. This is known as the **Twin Prime Conjecture**.