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# Divisor

For divisors in algebraic geometry, see divisor (algebraic geometry).

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 and we usually write 7 | 42. Divisors can be positive or negative. The positive divisors of 42 are {1, 2, 3, 6, 7, 14, 21, 42}.

Some special cases: 1 and -1 are divisors of every integer, and every integer is a divisor of 0. Numbers divisible by 2 are called even and those that are not are called odd.

The name comes from the arithmetic operation of division: if a/b=c then a is the dividend, b the divisor, and c the quotient.

 Contents

## Rules for small divisors

There are some rules which allow to recognize small divisors of a number from the number's decimal digits:

A divisibility rule is a rule you can use to determine a number's divisibility by another number. In decimal, the divisibility rules are:

• a number is divisible by 2 iff the last digit is divisible by 2
• a number is divisible by 3 iff the sum of its digits is divisible by 3
• a number is divisible by 4 iff the number given by the last two digits is divisible by 4
• a number is divisible by 5 iff the last digit is 0 or 5
• a number is divisible by 6 iff it is divisible by 2 and by 3
• a number is divisible by 7 iff the result of subtracting twice the last digit from the number with the last digit removed is divisible by 7 (e.g. 364 is divisible by 7 since 36-2×4 = 28 is divisible by 7)
• a number is divisible by 8 iff the number given by the last three digits is divisible by 8
• a number is divisible by 9 iff the sum of its digits is divisible by 9
• a number is divisible by 10 iff the last digit is 0
• a number is divisible by 11 iff the alternating sum of its digits is divisible by 11 (e.g. 182919 is divisible by 11 since 1-8+2-9+1-9 = -22 is divisible by 11)
• a number is divisible by 12 iff it's divisible by 3 and by 4
• A number is divisible by 13
• iff the result of adding 4 times the last digit to the original number.
• iff the result of subtracting 9 times the last digit from the number with the last digit removed is divisible by 13 (e.g. 858 is divisible by 13 since 85-9×8 = 13 is divisible by 13).
• A number is divisible by 14 iff it's divisible by 2 and by 7.
• A number is divisible by 15 iff it's divisible by 3 and by 5.

## Further notions and facts

Some elementary rules:

• If a | b and a | c, then a | (b + c).
• If a | b and b | c, then a | c.
• If a | b and b | a, then a = b or a = -b.

A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A number which does not evenly divide n, but leaves a remainder, is called an aliquant part of n.)

An integer n > 1 whose only proper divisor is 1 is called a prime number.

Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the Fundamental theorem of arithmetic.

If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than that sum are said to be deficient, while numbers greater than that sum are said to be abundant.

The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3×4×8 = σ(2)×σ(3)×σ(7)).

If the prime factorization of n is given by

$n = p_1^{\nu_1} \, p_2^{\nu_2} \, ... \, p_n^{\nu_n}$

then the number of positive divisors of n is

d(n) = (ν1 + 1)(ν2 + 1)...(νn + 1),

and each of the divisors has the form

$p_1^{\mu_1} \, p_2^{\mu_2} \, ... \, p_n^{\mu_n}$

where

$\forall i : 0 \le \mu_i \le \nu_i.$

The relation | of divisibility turns the set N 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 one 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 Z.

If an integer n is written in base b, and d is an integer with b ≡ 1 (mod d), then n is divisible by d if and only if the sum of its digits is divisible by d. The rules for d=3 and d=9 given above are special cases of this result (b=10).

## Generalization

One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting.