2. Composite Numbers
• composite number is a positive integer that has at least one positive divisor other
than one or itself. In other words, a composite number is any positive integer
greater than one that is not a prime number.[1][2]
• So, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b,
then n is composite. By definition, every integer greater than one is either a
prime number or a composite number. The number one is a unit;[3][4]
it is neither
prime nor composite. 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 first 105 composite numbers (sequence A002808 in OEIS) 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.Every composite number can be written as the product of two or more
(not necessarily distinct) primes;[5]
furthermore, this representation is unique up to
the order of the factors. This is called the fundamental theorem of arithmetic.
3. Prime Numbers
• A prime number (or a prime) is a natural number greater
than 1 that has no positive divisors other than 1 and itself.
A natural number greater than 1 that is not a prime
number is called a composite number. For example, 5 is
prime because only 1 and 5 evenly divide it, whereas 6 is
composite because it has the divisors 2 and 3 in addition to
1 and 6. The fundamental theorem of
arithmetic establishes the central role of primes in number
theory: any integer greater than 1 can be expressed as a
product of primes that is unique up toordering. The
uniqueness in this theorem requires excluding 1 as a prime
because one can include arbitrarily many copies of 1 in any
factorization, e.g., 3, 1 × 3, 1 × 1 × 3, etc. are all valid
factorizations of 3.
![Composite Numbers
• composite number is a positive integer that has at least one positive divisor other
than one or itself. In other words, a composite number is any positive integer
greater than one that is not a prime number.[1][2]
• So, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b,
then n is composite. By definition, every integer greater than one is either a
prime number or a composite number. The number one is a unit;[3][4]
it is neither
prime nor composite. 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 first 105 composite numbers (sequence A002808 in OEIS) 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.Every composite number can be written as the product of two or more
(not necessarily distinct) primes;[5]
furthermore, this representation is unique up to
the order of the factors. This is called the fundamental theorem of arithmetic.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)