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Composite number

Demonstration, with Cuisenaire rods, of the divisors of the composite number 10
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
Composite numbers 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.[1][2] Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit.[3][4] E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 –ut 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. (sequence A002808 in the OEIS)

Every composite number can be written as the product of two or more (not necessarily distinct) primes.[2] For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 23 × 32 × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic.[5][6][7][8]

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

  1. ^ Pettofrezzo & Byrkit 1970, pp. 23–24.
  2. ^ a b Long 1972, p. 16.
  3. ^ Fraleigh 1976, pp. 198, 266.
  4. ^ Herstein 1964, p. 106.
  5. ^ Fraleigh 1976, p. 270.
  6. ^ Long 1972, p. 44.
  7. ^ McCoy 1968, p. 85.
  8. ^ Pettofrezzo & Byrkit 1970, p. 53.

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