Divisor

An integer ${\displaystyle n}$  is divisible by a nonzero integer ${\displaystyle m}$  if there exists an integer ${\displaystyle k}$  such that ${\displaystyle n=km.}$  This is written as

${\displaystyle m\mid n.}$ 

This may be read as that ${\displaystyle m}$  divides ${\displaystyle n,}$  ${\displaystyle m}$  is a divisor of ${\displaystyle n,}$  ${\displaystyle m}$  is a factor of ${\displaystyle n,}$  or ${\displaystyle n}$  is a multiple of ${\displaystyle m.}$  If ${\displaystyle m}$  does not divide ${\displaystyle n,}$  then the notation is ${\displaystyle m\not \mid n.}$ ^[2][3]

There are two conventions, distinguished by whether ${\displaystyle m}$  is permitted to be zero:

• With the convention without an additional constraint on ${\displaystyle m,}$  ${\displaystyle m\mid 0}$  for every integer ${\displaystyle m.}$ ^[2][3]
• With the convention that ${\displaystyle m}$  be nonzero, ${\displaystyle m\mid 0}$  for every nonzero integer ${\displaystyle m.}$ ^[4][5]

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.

1, −1, ${\displaystyle n}$  and ${\displaystyle -n}$  are known as the trivial divisors of ${\displaystyle n.}$  A divisor of ${\displaystyle n}$  that is not a trivial divisor is known as a non-trivial divisor (or strict divisor^[6]). 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.

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

• 7 is a divisor of 42 because ${\displaystyle 7\times 6=42,}$  so we can say ${\displaystyle 7\mid 42.}$  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, ${\displaystyle A=\{1,2,3,4,5,6,10,12,15,20,30,60\},}$  partially ordered by divisibility, has the Hasse diagram:

There are some elementary rules:

• If ${\displaystyle a\mid b}$  and ${\displaystyle b\mid c,}$  then ${\displaystyle a\mid c;}$  that is, divisibility is a transitive relation.
• If ${\displaystyle a\mid b}$  and ${\displaystyle b\mid a,}$  then ${\displaystyle a=b}$  or ${\displaystyle a=-b.}$  (That is, ${\displaystyle a}$  and ${\displaystyle b}$  are associates.)
• If ${\displaystyle a\mid b}$  and ${\displaystyle a\mid c,}$  then ${\displaystyle a\mid (b+c)}$  holds, as does ${\displaystyle a\mid (b-c).}$ ^[a] However, if ${\displaystyle a\mid b}$  and ${\displaystyle c\mid b,}$  then ${\displaystyle (a+c)\mid b}$  does not always hold (for example, ${\displaystyle 2\mid 6}$  and ${\displaystyle 3\mid 6}$  but 5 does not divide 6).
• ${\displaystyle a\mid b\iff ac\mid bc}$  for nonzero ${\displaystyle c}$ . This follows immediately from writing ${\displaystyle ka=b\iff kac=bc}$ .

If ${\displaystyle a\mid bc,}$  and ${\displaystyle \gcd(a,b)=1,}$  then ${\displaystyle a\mid c.}$ ^[b] This is called Euclid's lemma.

If ${\displaystyle p}$  is a prime number and ${\displaystyle p\mid ab}$  then ${\displaystyle p\mid a}$  or ${\displaystyle p\mid b.}$ 

A positive divisor of ${\displaystyle n}$  that is different from ${\displaystyle n}$  is called a proper divisor or an aliquot part of ${\displaystyle n}$  (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide ${\displaystyle n}$  but leaves a remainder is sometimes called an aliquant part of ${\displaystyle n.}$ 

An integer ${\displaystyle n>1}$  whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.

Any positive divisor of ${\displaystyle n}$  is a product of prime divisors of ${\displaystyle n}$  raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number ${\displaystyle n}$  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 ${\displaystyle n,}$  and abundant if this sum exceeds ${\displaystyle n.}$ 

The total number of positive divisors of ${\displaystyle n}$  is a multiplicative function ${\displaystyle d(n),}$  meaning that when two numbers ${\displaystyle m}$  and ${\displaystyle n}$  are relatively prime, then ${\displaystyle d(mn)=d(m)\times d(n).}$  For instance, ${\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)}$ ; 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 ${\displaystyle m}$  and ${\displaystyle n}$  share a common divisor, then it might not be true that ${\displaystyle d(mn)=d(m)\times d(n).}$  The sum of the positive divisors of ${\displaystyle n}$  is another multiplicative function ${\displaystyle \sigma (n)}$  (for example, ${\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42}$ ). Both of these functions are examples of divisor functions.

If the prime factorization of ${\displaystyle n}$  is given by

${\displaystyle n=p_{1}^{\nu _{1}}\,p_{2}^{\nu _{2}}\cdots p_{k}^{\nu _{k}}}$ 

then the number of positive divisors of ${\displaystyle n}$  is

${\displaystyle d(n)=(\nu _{1}+1)(\nu _{2}+1)\cdots (\nu _{k}+1),}$ 

and each of the divisors has the form

${\displaystyle p_{1}^{\mu _{1}}\,p_{2}^{\mu _{2}}\cdots p_{k}^{\mu _{k}}}$ 

where ${\displaystyle 0\leq \mu _{i}\leq \nu _{i}}$  for each ${\displaystyle 1\leq i\leq k.}$ 

For every natural ${\displaystyle n,}$  ${\displaystyle d(n)<2{\sqrt {n}}.}$ 

Also,^[7]

${\displaystyle d(1)+d(2)+\cdots +d(n)=n\ln n+(2\gamma -1)n+O({\sqrt {n}}),}$ 

where ${\displaystyle \gamma }$  is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about ${\displaystyle \ln n.}$  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 ${\displaystyle \mathbb {N} }$  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.

• Arithmetic functions
• Euclidean algorithm
• Fraction (mathematics)
• Integer factorization
• Table of divisors – A table of prime and non-prime divisors for 1–1000
• Table of prime factors – A table of prime factors for 1–1000
• Unitary divisor