Divisible

"Divisible" redirects here. For divisibility of groups, see Divisible group.

For the second operand of a division, see Division (mathematics). For divisors in algebraic geometry, see Divisor (algebraic geometry). For divisibility in the ring theory, see Divisibility (ring theory).

Calculation results
Addition (+)
addend + addend =	sum
Subtraction (−)
minuend − subtrahend =	difference
Multiplication (×)
multiplicand × multiplier =	product
Division (÷)
dividend ÷ divisor =	quotient
Exponentiation
base =	power
nth root (√)
$degree \sqrt radicand$ =	root
Logarithm
log_base(power) =	exponent

The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

In mathematics, a divisor of an integer $n$ , also called a factor of $n$ , is an integer which divides $n$ without leaving a remainder.

Terminology [edit]

The name "divisor" comes from the arithmetic operation of division: if $\frac{a}{b} = c$ then $a$ is the dividend, $b$ the divisor, and $c$ the quotient.

In general, for non-zero integers $m$ and $n$ , it is said that $m$ divides $n$ -and, dually, that $n$ is divisible by $m$ -written:

$m \mid n,$

if there exists an integer $k$ such that $n = km$ . 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 and −n are known as the trivial divisors of n. 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.

Examples [edit]

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

Further notions and facts [edit]

There are some elementary rules:

If $a \mid b$ and $b \mid c$ , then $a \mid c$ . This is the transitive relation.
If $a \mid b$ and $b \mid a$ , then $a = b$ or $a = -b$ .
If $a \mid b$ and $c \mid b$ , then it is NOT always true that $(a + c) \mid b$ (e.g. $2\mid6$ and $3 \mid 6$ but 5 does not divide 6). However, when $a \mid b$ and $a \mid c$ , then $a \mid (b + c)$ is true, as is $a \mid (b - c)$ .

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

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

A positive divisor of $n$ which is different from $n$ is called a proper divisor or an aliquot part of $n$ . A number that 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. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.

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.

A number $n$ 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 $n$ , and abundant if this sum exceeds $n$ .

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

If the prime factorization of $n$ is given by

$n = p_1^{\nu_1} \, p_2^{\nu_2} \cdots p_k^{\nu_k}$

then the number of positive divisors of $n$ is

$d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_k + 1),$

and each of the divisors has the form

$p_1^{\mu_1} \, p_2^{\mu_2} \cdots p_k^{\mu_k}$

where $0 \le \mu_i \le \nu_i$ for each $1 \le i \le k.$

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

Also,

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

where $\gamma$ 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 $\ln n$ .

In abstract algebra [edit]

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 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 $Z$ .

Notes [edit]

Durbin, John R. (1992). Modern Algebra : an Introduction (3rd ed. ed.). New York: Wiley. p. 61. ISBN - get this book. "An integer $m$ is divisible by an integer $n$ if there is an integer $q$ (for quotient) such that $m = n q$ ."
$a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b+c=(j+k)a \Rightarrow a \mid (b+c)$ Similarly, $a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b-c=(j-k)a \Rightarrow a \mid (b-c)$
Hardy, G. H.; E. M. Wright (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264. ISBN - get this book.

References [edit]

Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 - get this book; section B.
Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).

v t e Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number

Constrained divisor sums	Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hemiperfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Practical number

With many divisors	Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number

Aliquot sequence-related	Untouchable number Amicable number Sociable number Betrothed number

Other sets	Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number

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