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. 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. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 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. (sequence A002808 in the OEIS)
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 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.
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.
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 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
(where μ is the Möbius function and x is half the total of prime factors), while for the former
However, for prime numbers, the function also returns −1 and 
. For a number n with one or more repeated prime factors,
![image]()
.
If all the prime factors of a number are repeated it is called a powerful number (All perfect powers are powerful numbers). If none of its prime factors are repeated, it is called squarefree. (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 (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 numbers, 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 numbers and rough numbers, respectively.
See also
- Canonical representation of a positive integer
- Integer factorization
- Sieve of Eratosthenes
- Table of prime factors
Notes
- Pettofrezzo & Byrkit 1970, pp. 23–24.
- Long 1972, p. 16.
- Fraleigh 1976, pp. 198, 266.
- Herstein 1964, p. 106.
- Fraleigh 1976, p. 270.
- Long 1972, p. 44.
- McCoy 1968, p. 85.
- Pettofrezzo & Byrkit 1970, p. 53.
- Long 1972, p. 159.
References
- Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
- Herstein, I. N. (1964), Topics In Algebra, Waltham: , ISBN 978-1114541016
- Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
- McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225
- Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
External links
- Lists of composites with prime factorization (first 100, 1,000, 10,000, 100,000, and 1,000,000)
- Divisor Plot (patterns found in large composite numbers)
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 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 E g the integer 14 is a composite number because it is the product of the two smaller integers 2 7 but the integers 2 and 3 are not because each can only be divided by one and itself Demonstration with Cuisenaire rods of the divisors of the composite number 10Composite numbers can be arranged into rectangles but prime numbers cannot 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 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 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 TypesOne 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 m n 1 2x 1 displaystyle mu n 1 2x 1 where m is the Mobius function and x is half the total of prime factors while for the former m n 1 2x 1 1 displaystyle mu n 1 2x 1 1 However for prime numbers the function also returns 1 and m 1 1 displaystyle mu 1 1 For a number n with one or more repeated prime factors m n 0 displaystyle mu n 0 If all the prime factors of a number are repeated it is called a powerful number All perfect powers are powerful numbers If none of its prime factors are repeated it is called squarefree 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 Euler diagram of numbers under 100 Abundant Primitive abundant Highly abundant Superabundant and highly composite Colossally abundant and superior highly composite Weird Perfect Composite Deficient 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 1 p p2 displaystyle 1 p p 2 A number n that has more divisors than any x lt n is a highly composite number 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 numbers 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 numbers and rough numbers respectively See alsoMathematics portalCanonical representation of a positive integer Integer factorization Sieve of Eratosthenes Table of prime factorsNotesPettofrezzo amp Byrkit 1970 pp 23 24 Long 1972 p 16 Fraleigh 1976 pp 198 266 Herstein 1964 p 106 Fraleigh 1976 p 270 Long 1972 p 44 McCoy 1968 p 85 Pettofrezzo amp Byrkit 1970 p 53 Long 1972 p 159 ReferencesFraleigh John B 1976 A First Course In Abstract Algebra 2nd ed Reading Addison Wesley ISBN 0 201 01984 1 Herstein I N 1964 Topics In Algebra Waltham ISBN 978 1114541016 Long Calvin T 1972 Elementary Introduction to Number Theory 2nd ed Lexington D C Heath and Company LCCN 77 171950 McCoy Neal H 1968 Introduction To Modern Algebra Revised Edition Boston Allyn and Bacon LCCN 68 15225 Pettofrezzo Anthony J Byrkit Donald R 1970 Elements of Number Theory Englewood Cliffs Prentice Hall LCCN 77 81766External linksLists of composites with prime factorization first 100 1 000 10 000 100 000 and 1 000 000 Divisor Plot patterns found in large composite numbers
