What Is A Divisor In Math

A divisor in math is the number you are dividing with in an equation. EX: 20/4=5 The dividend is 20 The divisor is 4 The quotient is 5 20 divided by 4 equals 5. Define divisor. Divisor synonyms, divisor pronunciation, divisor translation, English dictionary definition of divisor. The quantity by which another quantity, the dividend, is to be divided. A number or quantity to be divided into another number or quantity 2. Divide, division, dividend, divisor. to divide or division is sharing or grouping. A number into equal parts. dividend: the number being divided. divisor or factor: a number that. Will divide the dividend exactly. The divisor is any number that divides another number. A factor, however, is a divisor that divides the number entirely and leaves no remainder. So, all factors of a number are its divisors. In mathematics, a divisor of an integer, also called a factor of, is an integer that may be multiplied by some integer to produce. In this case, one also says that is a multiple of An integer is divisible by another integer if is a divisor of; this implies dividing by leaves no remainder.

1. What Is A Divisor In Math Problem
2. What Is The Meaning Of Divisor In Mathematics
3. What Does A Divisor In Math

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

Noun Mathematics. A number by which another number, the dividend, is divided. A number contained in another given number a certain integral number of times, without a remainder. Origin of divisor.

Definition[edit]

If ${\displaystyle m}$ and ${\displaystyle n}$ are nonzero integers, and more generally, nonzero elements of an integral domain, it is said that ${\displaystyle m}$divides${\displaystyle n}$, ${\displaystyle m}$ is a divisor of ${\displaystyle n,}$ or ${\displaystyle n}$ is a multiple of ${\displaystyle m,}$ and this is written as

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${\displaystyle m\mid n,}$

if there exists an integer ${\displaystyle k}$, or an element ${\displaystyle k}$ of the integral domain, such that ${\displaystyle mk=n}$.^[1]

This definition is sometimes extended to include zero.^[2] This does not add much to the theory, as 0 does not divide any other number, and every number divides 0. On the other hand, excluding zero from the definition simplifies many statements. Also, in ring theory, an element a is called a 'zero divisor' only if it is nonzero and ab = 0 for a nonzero element b. Thus, there are no zero divisors among the integers (and by definition no zero divisors in an integral domain).

General[edit]

Divisors can be negative as well as positive, although sometimes 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 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 ( or strict divisor ^[3]) . A non-zero 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[edit]

• 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:

Further notions and facts[edit]

There are some elementary rules:

• If ${\displaystyle a\mid b}$ and ${\displaystyle b\mid c}$, then ${\displaystyle a\mid c}$, i.e. divisibility is a transitive relation.
• If ${\displaystyle a\mid b}$ and ${\displaystyle b\mid a}$, then ${\displaystyle a=b}$ or ${\displaystyle a=-b}$.
• If ${\displaystyle a\mid b}$ and ${\displaystyle a\mid c}$, then ${\displaystyle a\mid (b+c)}$ holds, as does ${\displaystyle a\mid (b-c)}$.^[4] However, if ${\displaystyle a\mid b}$ and ${\displaystyle c\mid b}$, then ${\displaystyle (a+c)\mid b}$ does not always hold (e.g. ${\displaystyle 2\mid 6}$ and ${\displaystyle 3\mid 6}$ but 5 does not divide 6).

What Is A Divisor In Math Problem

If ${\displaystyle a\mid bc}$, and gcd${\displaystyle (a,b)=1}$, then ${\displaystyle a\mid c}$. 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 ${\displaystyle n}$ which is different from ${\displaystyle n}$ is called a proper divisor or an aliquot part of ${\displaystyle n}$. A number that does not evenly divide ${\displaystyle n}$ but leaves a remainder is 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 ${\displaystyle \sigma (n)}$ (e.g. ${\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42}$). 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 ${\displaystyle 0\le {\mu }_i\le {\nu }_i}$ for each ${\displaystyle 1\le i\le k.}$

For every natural ${\displaystyle n}$, .

Also,^[5]

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

where ${\displaystyle \gamma }$ is Euler–Mascheroni constant.One interpretation of this result is that a randomly chosen positive integer n has an averagenumber of divisors of about ${\displaystyle \ln n}$. However, this is a result from the contributions of small and 'abnormally large' divisors.

What Is The Meaning Of Divisor In Mathematics

In abstract algebra[edit]

In definitions that include 0, the relation of divisibility turns the set ${\displaystyle \mathbb{N}}$ of non-negative integers into a partially ordered set: 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${\displaystyle \mathbb{Z}}$.

See also[edit]

What Does A Divisor In Math

• 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

Notes[edit]

1. ^for instance, Sims 1984, p. 42 or Durbin 1992, p. 61
2. ^Herstein 1986, p. 26
3. ^FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois
4. ^${\displaystyle a\mid b,a\mid c\Rightarrow b=ja,c=ka\Rightarrow b+c=(j+k)a\Rightarrow a\mid (b+c)}$. Similarly, ${\displaystyle a\mid b,a\mid c\Rightarrow b=ja,c=ka\Rightarrow b-c=(j-k)a\Rightarrow a\mid (b-c)}$
5. ^Hardy, G. H.; Wright, E. M. (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264. ISBN0-19-853171-0.

References[edit]

• Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN0-471-51001-7.
• Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN0-387-20860-7; section B.
• Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN0-02-353820-1
• Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
• Sims, Charles C. (1984), Abstract Algebra: A Computational Approach, New York: John Wiley & Sons, ISBN0-471-09846-9