Composite number
A composite number is a positive integer that has at least one positive divisor other than one or the number itself. Any integer greater than one that is not a prime number is a composite number,[1][2] while the number one is a unit; it is neither prime nor composite.[3][4]
For example, the integer 14 is a composite number because it can be factored as 2 × 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. (sequence A002808 in OEIS)
Every composite number can be written as the product of two or more (not necessarily distinct) primes.[5] 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.[6][7][8][9]
There are several known primality tests 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 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,
.[10]
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).
Factorization
Here are all the composite numbers less than or equal to 150 and their factorization:
4 = 22
6 = 2 × 3
8 = 23
9 = 32
10 = 2 × 5
12 = 22 × 3
14 = 2 × 7
15 = 3 × 5
16 = 24
18 = 2 × 32
20 = 22 × 5
21 = 3 × 7
22 = 2 × 11
24 = 23 × 3
25 = 52
26 = 2 × 13
27 = 33
28 = 22 × 7
30 = 2 × 3 × 5
32 = 25
33 = 3 × 11
34 = 2 × 17
35 = 5 × 7
36 = 22 × 32
38 = 2 × 19
39 = 3 × 13
40 = 23 × 5
42 = 2 × 3 × 7
44 = 22 × 11
45 = 32 × 5
46 = 2 × 23
48 = 24 × 3
49 = 72
50 = 2 × 52
51 = 3 × 17
52 = 22 × 13
54 = 2 × 33
55 = 5 × 11
56 = 23 × 7
57 = 3 × 19
58 = 2 × 29
60 = 22 × 3 × 5
62 = 2 × 31
63 = 32 × 7
64 = 26
65 = 5 × 13
66 = 2 × 3 × 11
68 = 22 × 17
69 = 3 × 23
70 = 2 × 5 × 7
72 = 23 × 32
74 = 2 × 37
75 = 3 × 52
76 = 22 × 19
77 = 7 × 11
78 = 2 × 3 × 13
80 = 24 × 5
81 = 34
82 = 2 × 41
84 = 22 × 3 × 7
85 = 5 × 17
86 = 2 × 43
87 = 3 × 29
88 = 23 × 11
90 = 2 × 32 × 5
91 = 7 × 13
92 = 22 × 23
93 = 3 × 31
94 = 2 × 47
95 = 5 × 19
96 = 25 × 3
98 = 2 × 72
99 = 32 × 11
100 = 22 × 52
102 = 2 × 3 × 17
104 = 23 × 13
105 = 3 × 5 × 7
106 = 2 × 53
108 = 22 × 33
110 = 2 × 5 × 11
111 = 3 × 37
112 = 24 × 7
114 = 2 × 3 × 19
115 = 5 × 23
116 = 22 × 29
117 = 32 × 13
118 = 2 × 59
119 = 7 × 17
120 = 23 × 3 × 5
121 = 112
122 = 2 × 61
123 = 3 × 41
124 = 4 × 31
125 = 53
126 = 2 × 32 × 7
128 = 27
129 = 3 × 43
130 = 2 × 5 × 13
132 = 22 × 3 × 11
133 = 7 × 19
134 = 2 × 67
135 = 33 × 5
136 = 23 × 17
138 = 2 × 3 × 23
140 = 22 × 5 × 7
141 = 3 × 47
142 = 2 × 71
143 = 11 × 13
144 = 24 × 32
145 = 5 × 29
146 = 2 × 73
147 = 3 × 72
148 = 22 × 37
150 = 2 × 3 × 52
See also
- Table of prime factors
- Integer factorization
- Canonical representation of a positive integer
- Sieve of Eratosthenes
Notes
<templatestyles src="Reflist/styles.css" />
Cite error: Invalid <references> tag; parameter "group" is allowed only.
<references />, or <references group="..." />References
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
External links
- An integer factorizer, can factor all integers less than 1060
- Java applet: Factorization using the Elliptic Curve Method to find very large composites
- 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)
|
Divisibility-based sets of integers
|
||
|---|---|---|
| Overview |
 |
|
| Factorization forms | ||
| Constrained divisor sums | ||
| With many divisors | ||
| Aliquot sequence-related | ||
| Other sets | ||
- ↑ Pettofrezzo (1970, pp. 23–24)
- ↑ Long (1972, p. 16)
- ↑ Fraleigh (1976, pp. 198,266)
- ↑ Herstein (1964, p. 106)
- ↑ Long (1972, p. 16)
- ↑ Fraleigh (1976, p. 270)
- ↑ Long (1972, p. 44)
- ↑ McCoy (1968, p. 85)
- ↑ Pettofrezzo (1970, p. 53)
- ↑ Long (1972, p. 159)
