Composite number

Every composite number can be written as the product of 2 or more (not necessarily distinct) primes (Fundamental theorem of arithmetic).
Also, Failed to parse (Missing texvc executable; please see math/README to configure.): (n-1)! \,\,\, \equiv \,\, 0 \pmod{n}

 for all composite numbers n > 5.  See also Wilson's theorem.

Kinds of composite numbers

One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(n) = (-1)^{2x} = 1\,

(where μ is the Möbius function and x is half the total of prime factors), while for the former

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(n) = (-1)^{2x + 1} = -1.\,

Note however that for prime numbers the function also returns -1, and that Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(1) = 1 . For a number n with one or more repeated prime factors, Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(n) = 0 .

Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are Failed to parse (Missing texvc executable; please see math/README to configure.): \{1, p, p^2\} . A number n that has more divisors than any x < n is a highly composite number (though the first two such numbers are 1 and 2).