What is a Composite Number?
A composite number is a positive integer greater than 1 that has at least one positive divisor other than 1 and itself. In other words, a composite number can be formed by multiplying two smaller positive integers. The first few composite numbers are 4, 6, 8, 9, 10, 12, 14, and 15.
Every positive integer greater than 1 is either prime or composite. The numbers 0 and 1 are neither. Composite numbers are the “opposite” of primes — while primes are indivisible, composites can always be broken down into smaller factors. The Fundamental Theorem of Arithmetic guarantees that every composite number has a unique prime factorization.
Properties of Composite Numbers
Every composite number has at least three divisors: 1, itself, and at least one other divisor. The smallest composite number is 4 = 2 × 2. The number of divisors a composite number has depends on its prime factorization — numbers with many small prime factors tend to have many divisors. For example, 60 = 2² × 3 × 5 has 12 divisors, while 64 = 26 has only 7.
Composite numbers are far more common than primes as numbers grow larger. Among the first 100 positive integers, 74 are composite and 25 are prime (plus the number 1). Among the first million, about 92% are composite. Understanding the structure of composite numbers through their prime factorization is fundamental to cryptography, coding theory, and computational number theory.
Highly Composite Numbers
A highly composite number is a positive integer with more divisors than any smaller positive integer. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360... These numbers were studied extensively by Srinivasa Ramanujan in 1915 and are important in applications where having many divisors is desirable, such as choosing the number of hours in a day (24) or degrees in a circle (360).