Composite Numbers

A composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself. In other words, it can be formed by multiplying two smaller positive integers. For example, 12 is composite because it equals 2 × 6, 3 × 4 or 2 × 2 × 3. Every positive integer greater than 1 is either prime or composite — these two categories together cover all integers from 2 onwards.

What are composite numbers?

A composite number is any integer greater than 1 that is not prime. While a prime number has exactly two distinct positive divisors (1 and itself), a composite number has three or more. The smallest composite number is 4 (divisors: 1, 2, 4), followed by 6, 8, 9, 10, 12, 14, 15... Note that 1 is neither prime nor composite — it occupies a special category of its own called a unit. Composite numbers are far more common than primes: among the first 100 positive integers, there are 74 composites but only 25 primes.

Properties of composite numbers

Composite numbers have several important mathematical properties. By the Fundamental Theorem of Arithmetic, every composite number can be expressed as a unique product of prime factors (up to the order of the factors). For instance, 60 = 2² × 3 × 5. This factorization is central to number theory, cryptography and computer science. Every composite number has at least three divisors. The number of divisors of a composite is determined by its prime factorization: if n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, then the total number of divisors is (a₁+1)(a₂+1)...(aₖ+1). Composite numbers also appear in the Sieve of Eratosthenes — they are precisely the numbers that get eliminated during the sieving process.

Relationship with prime numbers

Composite numbers and prime numbers are complementary sets: every integer greater than 1 belongs to exactly one of the two categories. While primes are the "atoms" of arithmetic, composites are the "molecules" — built from prime factors. The Prime Number Theorem tells us that the density of primes decreases as numbers grow larger, which means composite numbers become proportionally more common. Among the first 10 integers (2-10), about 44% are composite; among the first 1,000,000, approximately 92% are composite. Every even number greater than 2 is composite, since it is divisible by 2. Among odd numbers, composites are also abundant: 9, 15, 21, 25, 27, 33, 35, 39...

Types of composite numbers

Mathematicians have identified several interesting subclasses of composite numbers:

• Highly composite numbers: Numbers that have more divisors than any smaller positive integer. The sequence begins 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360... These were studied extensively by Ramanujan in 1915 and are important in optimization problems.
• Semiprimes: Products of exactly two primes (not necessarily distinct), such as 4 = 2×2, 6 = 2×3, 9 = 3×3, 10 = 2×5, 14 = 2×7, 15 = 3×5. Semiprimes are central to RSA cryptography, where the security depends on the difficulty of factoring the product of two large primes.
• Smooth numbers: Numbers whose prime factors are all small. A number is B-smooth if all its prime factors are ≤ B. For example, 12 = 2² × 3 is 3-smooth. Smooth numbers are important in integer factorization algorithms such as the quadratic sieve and the number field sieve.
• Perfect powers: Composites that are exact powers of a smaller integer, like 4 = 2², 8 = 2³, 9 = 3², 16 = 2⁴, 25 = 5², 27 = 3³. These form a sparse but well-studied subset of composites.
• Pseudoprimes: Composite numbers that pass certain primality tests designed to detect primes. Carmichael numbers (561, 1105, 1729...) are composites that pass the Fermat primality test for every base, making them particularly tricky to identify.

Did you know?

• The number 4 is the smallest composite number. It is the only composite that equals 2² and is also a perfect square.
• Every even number greater than 2 is composite, since it is divisible by 2. This makes 2 the only even prime number.
• The number 1 is neither prime nor composite. It was considered prime until the early 20th century, but modern convention excludes it to preserve the uniqueness of prime factorization.
• Highly composite numbers like 12, 24, 60 and 360 were favoured by ancient civilizations for measurement systems — this is why we have 12 hours, 24 hours in a day, 60 minutes in an hour and 360 degrees in a circle.
• The product of any two integers greater than 1 is always composite. This means composite numbers are "closed under multiplication" in a sense: multiplying composites (or primes) always produces a composite.

List of the first 50 composite numbers

Click on any composite number to see its full analysis with prime factorization, divisors, conversions and curiosities.