Divisor

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In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.

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Explanation

The name "divisor" comes from the arithmetic operation of division: if a / b = c then a is the dividend, b the divisor, and c the quotient.

In general, for non-zero integers m and n, m divides n, written:

if there exists an integer k such that n = km. Thus, divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.)

1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non-trivial divisor. A number with at least one non-trivial divisor is known as a composite number, while the units -1 and 1 and prime numbers have no non-trivial divisors.

There are divisibility rules which allow one to recognize certain divisors of a number from the number's digits.

Examples

  • 7 is a divisor of 42 because 42 / 7 = 6, so we can say 7 \mid 42. It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42.
  • The non-trivial divisors of 6 are 2, −2, 3, −3.
  • The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
  • The set of all positive divisors of 60, A = {1,2,3,4,5,6,10,12,15,20,30,60}, partially ordered by divisibility, has the Hasse diagram:

Further notions and facts

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