Divisor

In mathematics, a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,} is an integer m {\displaystyle m} that may be multiplied by some integer to produce n . {\displaystyle n.} In this case, one also says that n {\displaystyle n} is a multiple of m . {\displaystyle m.} An integer n {\displaystyle n} is divisible or evenly divisible by another integer m {\displaystyle m} if m {\displaystyle m} is a divisor of n {\displaystyle n}; this implies dividing n {\displaystyle n} by m {\displaystyle m} leaves no remainder.

The concept of a divisor is extended, with the same definition, to elements of any ring; see Divisibility (ring theory).

Definition

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

m ∣ n . {\displaystyle m\mid n.}

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

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

With the convention without an additional constraint on m , {\displaystyle m,} m ∣ 0 {\displaystyle m\mid 0} for every integer m . {\displaystyle m.}
With the convention that m {\displaystyle m} be nonzero, m ∣ 0 {\displaystyle m\mid 0} for every nonzero integer m . {\displaystyle m.}

General

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

Examples

7 is a divisor of 42 because 7 × 6 = 42 , {\displaystyle 7\times 6=42,} so we can say 7 ∣ 42. {\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, A = { 1 , 2 , 3 , 4 , 5 , 6 , 10 , 12 , 15 , 20 , 30 , 60 } , {\displaystyle 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

There are some elementary rules:

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

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

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

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

An integer n > 1 {\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 n {\displaystyle n} is a product of prime divisors of n {\displaystyle n}, each raised to some power. This is a consequence of the fundamental theorem of arithmetic.

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

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

n = p 1 ν 1 p 2 ν 2 ⋯ p k ν k {\displaystyle n=p_{1}^{\nu _{1}}\,p_{2}^{\nu _{2}}\cdots p_{k}^{\nu _{k}}}

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

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

and each of the divisors has the form

p 1 μ 1 p 2 μ 2 ⋯ p k μ k {\displaystyle p_{1}^{\mu _{1}}\,p_{2}^{\mu _{2}}\cdots p_{k}^{\mu _{k}}}

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

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

Also,

d ( 1 ) + d ( 2 ) + ⋯ + d ( n ) = n ln ⁡ n + ( 2 γ − 1 ) n + O ( n ) , {\displaystyle d(1)+d(2)+\cdots +d(n)=n\ln n+(2\gamma -1)n+O({\sqrt {n}}),}

where γ {\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 ln ⁡ n . {\displaystyle \ln n.} However, this is a result from the contributions of numbers with "abnormally many" divisors.

Division lattice

In definitions that allow the divisor to be 0, the relation of divisibility turns the set N {\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.

Notes

Citations

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Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5thed.). John Wiley & Sons. ISBN0-471-62546-9.
Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
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