Divisor

In mathematics, a divisor of an integer

that may be multiplied by some integer to produce

is divisible by a nonzero integer

is permitted to be zero: Divisors can be negative as well as positive, although often the term is restricted to positive divisors.

For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.

1 and −1 divide (are divisors of) every integer.

Every integer (and its negation) is a divisor of itself.

Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.

A nonzero integer 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.

[b] This is called Euclid's lemma.

is called a proper divisor or an aliquot part of

(for example, the proper divisors of 6 are 1, 2, and 3).

A number that does not evenly divide

but leaves a remainder is sometimes called an aliquant part of

whose only proper divisor is 1 is called a prime number.

is a product of prime divisors of

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

share a common divisor, then it might not be true that

The sum of the positive divisors of

is given by then the number of positive divisors of

One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about

However, this is a result from the contributions of numbers with "abnormally many" divisors.

In definitions that allow the divisor to be 0, the relation of divisibility turns the set

of non-negative integers into a partially ordered set that is 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 ∨ by the least common multiple.

This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.