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A divisor is a number that divides evenly into another number without leaving any remainder. If n is a multiple of m, then m is called a divisor of n. A number n is divisible by m if and only if n == 0 (mod m).

Definition[]

If m and n are positive integers and m is a divisor of n, then m divides n and n is a multiple of m. More generally, it is said that n | m.

Sometimes divided by zero is included in this definition. This does not add much to the theory, because 0 does not divide any number except zero itself. In ring theory, a is called a zero divisor only if it is nonzero and if ab=0. Therefore, no nonzero integer is divisible by zero.

General[]

Divisors can be positive as well as negative, but the term "divisor" usually refers to only positive divisors. For example, 10 has 6 divisors, namely 1, 2, 3, 4, 6 and 10, or 10 divisors when you include -1, -2, -3 −4, −6 and -10.

1 and -1 divide every integer. Every integer n is divisible by the number n itself and -n. Integers which have 2 and -2 as divisors are even numbers, while numbers without 2 or -2 as divisors are odd numbers.

A nontrivial divisor of n is a number which is a divisor of n other than 1, -1, n, and -n. A nonzero integer with at least one nontrivial divisor is known as a composite number. The prime numbers, 1 and -1 have no nontrivial divisors.