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A composite number is a
positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

divisor
other than 1 and itself. Every positive integer is composite,
prime A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, or the
unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer
14
14
is a composite number because it is the product of the two smaller integers
2
2
 × 
7
7
. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150. Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example, the composite number 299 can be written as 13 × 23, and the composite number
360 360 may refer to: * 360 (number) 360 (three hundred sixty) is the natural number following 359 (number), 359 and preceding 361 (number), 361. In mathematics *The divisors of 360 are 1 (number), 1, 2 (number), 2, 3 (number), 3, 4 (number), 4, 5 ( ...
can be written as 23 × 32 × 5; furthermore, this representation is unique
up to Two mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is a ...
the order of the factors. This fact is called the
fundamental theorem of arithmetic In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
. There are several known
primality test A primality test is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specification ...
s that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.


Types

One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a
semiprime In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
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 numberIn number theory, a sphenic number (from grc, σφήνα, 'wedge') is a positive integer that is the product of three distinct prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), pr ...
. 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 :\mu(n) = (-1)^ = 1 (where μ is the
Möbius function The Möbius function is an important multiplicative function :''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' ...
and ''x'' is half the total of prime factors), while for the former :\mu(n) = (-1)^ = -1. However, for prime numbers, the function also returns −1 and \mu(1) = 1. For a number ''n'' with one or more repeated prime factors, :\mu(n) = 0. If ''all'' the prime factors of a number are repeated it is called a
powerful number A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a Square number, square and a Cube (arithmetic), cube, that is, a numb ...
(All
perfect power In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are powerful numbers). If ''none'' of its prime factors are repeated, it is called
squarefree In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. (All prime numbers and 1 are squarefree.) For example, 72 = 23 × 32, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. 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 \. A number ''n'' that has more divisors than any ''x'' < ''n'' is a
highly composite number __FORCETOC__ A highly composite number (sometimes called ''anti-prime'') is a positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Converging lens or positive lens, in optics * ...
(though the first two such numbers are 1 and 2). Composite numbers have also been called "rectangular numbers", but that name can also refer to the
pronic number A pronic number is a number which is the product of two consecutive integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoke ...
s, numbers that are the product of two consecutive integers. Yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed (prime) number. Such numbers are called
smooth number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number th ...
s and
rough numberA ''k''-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to ''k''. ''k''-roughness has alternately been defined as requiring all prime factors to strictly exceed ''k''.p. 130, ...
s, respectively.


See also

*
Canonical representation of a positive integer In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
*
Integer factorization In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
*
Sieve of Eratosthenes In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

Sieve of Eratosthenes
*
Table of prime factors The tables contain the integer factorization, prime factorization of the natural numbers from 1 to 1000. When ''n'' is a prime number, the prime factorization is just ''n'' itself, written in bold below. The number 1 (number), 1 is called a unit ( ...


Notes


References

* * * * *


External links


Lists of composites with prime factorization (first 100, 1,000, 10,000, 100,000, and 1,000,000)


{{Divisor classes Composite Integer sequences Arithmetic Elementary number theory