Editing Composite number (section)

{{Short description|Integer having a non-trivial divisor}}
[[File:Composite number Cuisenaire rods 10.svg|thumb|Demonstration, with [[Cuisenaire rods]], of the divisors of the composite number 10]]
[[File:Primes-vs-composites.svg | thumb|right | Composite numbers can be arranged into [[rectangles]] but prime numbers cannot.|alt=Groups of two to twelve dots, showing that the composite numbers of dots (4, 6, 8, 9, 10, and 12) can be arranged into rectangles but prime numbers cannot]]

A '''composite number''' is a [[positive integer]] that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one [[divisor]] other than 1 and itself.{{sfn|Pettofrezzo|Byrkit|1970|pp=23–24}}{{sfn|Long|1972|p=16}} Every positive integer is composite, [[prime number|prime]], or the [[Unit (ring theory)|unit]]&nbsp;1, so the composite numbers are exactly the numbers that are not prime and not a unit.{{sfn|Fraleigh|1976|pp=198,266}}{{sfn|Herstein|1964|p=106}} E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2&nbsp;&times;&nbsp;7 but the integers 2 and 3 are not because each 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. {{OEIS|id=A002808}}

Every composite number can be written as the product of two or more (not necessarily distinct) primes.{{sfn|Long|1972|p=16}} For example, the composite number [[299 (number)|299]] can be written as 13 × 23, and the composite number [[360 (number)|360]] can be written as 2<sup>3</sup> × 3<sup>2</sup> × 5; furthermore, this representation is unique [[up to]] the order of the factors. This fact is called the [[fundamental theorem of arithmetic]].{{sfn|Fraleigh|1976|p=270}}{{sfn|Long|1972|p=44}}{{sfn|McCoy|1968|p=85}}{{sfn|Pettofrezzo|Byrkit|1970|p=53}}

There are several known [[primality test]]s that can determine whether a number is prime or composite which do not necessarily reveal the [[factorization]] of a composite input.