In mathematics, a divisor
of an integer
, also called a factor
, is an integer which divides
without leaving a remainder.
The name "divisor" comes from the arithmetic operation of division: if
is the dividend,
In general, for non-zero integers
, it is said that divides
—and, dually, that
if there exists an integer
. 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
are known as the trivial divisors
. 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.
The generalization can be said to be the concept of divisibility
in any integral domain.
There are some elementary rules:
, and gcd
. This is called Euclid's lemma.
is a prime number and
A positive divisor of
which is different from
is called a proper divisor
or an aliquot part
. A number that does not evenly divide
but leaves a remainder is called an aliquant part
whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.
Any positive divisor of
is a product of prime divisors of
raised to some power. This is a consequence of the fundamental theorem of arithmetic.
is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than
, and abundant if this sum exceeds
The total number of positive divisors of
is a multiplicative function
, meaning that when two numbers
are relatively prime, then
. For instance,
; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers
share a common divisor, then it might not be true that
. The sum of the positive divisors of
is another multiplicative function
). Both of these functions are examples of divisor functions.
If the prime factorization of
is given by
then the number of positive divisors of
and each of the divisors has the form
For every natural
is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n
has an expected number of divisors of about
The relation of divisibility turns the set
of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group