What Is A Proper Factor

Integer that is a factor of another integer

This commodity is about an integer that is a cistron of another integer. For a number used to divide some other number in a division functioning, see Division (mathematics). For other uses, come across Divisor (disambiguation).

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

In mathematics, a divisor of an integer ${\displaystyle n}$ , besides called a factor of ${\displaystyle n}$ , is an integer ${\displaystyle thou}$ that may exist multiplied by some integer to produce ${\displaystyle n}$ . In this case, 1 also says that ${\displaystyle n}$ is a multiple of ${\displaystyle yard.}$ An integer ${\displaystyle n}$ is divisible or evenly divisible by another integer ${\displaystyle m}$ if ${\displaystyle m}$ is a divisor of ${\displaystyle north}$ ; this implies dividing ${\displaystyle n}$ by ${\displaystyle thousand}$ leaves no residual.

Definition [edit]

An integer n is divisible past a nonzero integer thousand if in that location exists an integer grand such that ${\displaystyle north=km}$ . This is written every bit

${\displaystyle m\mid n.}$

Other ways of saying the aforementioned thing are that m divides n, thou is a divisor of north, m is a factor of n, and northward is a multiple of m. If thousand does not divide due north, then the notation is ${\displaystyle one thousand\not \mid n}$ .^{[1] [2]}

Usually, thou is required to be nonzero, but due north is immune to be zero. With this convention, ${\displaystyle chiliad\mid 0}$ for every nonzero integer m.^{[1] [2]} Some definitions omit the requirement that ${\displaystyle one thousand}$ be nonzero.^[iii]

General [edit]

Divisors can be negative as well every bit positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, iv, −one, −2, and −4, simply simply the positive ones (i, 2, and 4) would usually exist mentioned.

1 and −one split up (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.

ane, −one, n and −due north are known equally the picayune divisors of due north. A divisor of due north that is not a trivial divisor is known as a not-trivial divisor (or strict divisor^[iv]). A nonzero integer with at least one non-niggling divisor is known equally a blended number, while the units −1 and one and prime number numbers have no non-trivial divisors.

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

Examples [edit]

• seven is a divisor of 42 because ${\displaystyle 7\times half dozen=42}$ , so we tin can say ${\displaystyle 7\mid 42}$ . It can likewise be said that 42 is divisible by 7, 42 is a multiple of seven, vii divides 42, or 7 is a factor of 42.
• The non-trivial divisors of 6 are two, −ii, 3, −iii.
• The positive divisors of 42 are one, 2, 3, 6, 7, 14, 21, 42.
• The fix of all positive divisors of threescore, ${\displaystyle A=\{one,ii,3,4,5,6,10,12,15,xx,30,60\}}$ , partially ordered by divisibility, has the Hasse diagram:

Further notions and facts [edit]

At that place are some elementary rules:

If ${\displaystyle a\mid bc}$ , and ${\displaystyle \gcd(a,b)=1}$ , so ${\displaystyle a\mid c}$ .^{[annotation ane]} This is called Euclid'south 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 ${\displaystyle northward}$ is called a proper divisor or an aliquot part of ${\displaystyle north}$ . A number that does not evenly divide ${\displaystyle n}$ simply leaves a balance is sometimes called an aliquant part of ${\displaystyle due north}$ .

An integer ${\displaystyle n>1}$ whose just proper divisor is 1 is called a prime. Equivalently, a prime number number is a positive integer that has exactly 2 positive factors: one 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 arithmetics.

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 arable if this sum exceeds ${\displaystyle n}$ .

The total number of positive divisors of ${\displaystyle n}$ is a multiplicative function ${\displaystyle d(north)}$ , significant that when two numbers ${\displaystyle m}$ and ${\displaystyle n}$ are relatively prime, then ${\displaystyle d(mn)=d(m)\times d(n)}$ . For instance, ${\displaystyle d(42)=8=2\times two\times 2=d(2)\times d(3)\times d(seven)}$ ; the eight divisors of 42 are 1, two, 3, vi, seven, 14, 21 and 42. Notwithstanding, the number of positive divisors is not a totally multiplicative function: if the two numbers ${\displaystyle thou}$ and ${\displaystyle n}$ share a common divisor, then it might not be true that ${\displaystyle d(mn)=d(m)\times d(northward)}$ . The sum of the positive divisors of ${\displaystyle n}$ is another multiplicative function ${\displaystyle \sigma (due north)}$ (e.g. ${\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=i+2+3+half-dozen+vii+fourteen+21+42}$ ). Both of these functions are examples of divisor functions.

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

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

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

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

and each of the divisors has the form

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

where ${\displaystyle 0\leq \mu _{i}\leq \nu _{i}}$ for each ${\displaystyle one\leq i\leq chiliad.}$

For every natural ${\displaystyle due north}$ , ${\displaystyle d(northward)<two{\sqrt {n}}}$ .

Also,^[6]

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

where ${\displaystyle \gamma }$ is Euler–Mascheroni constant. 1 interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of virtually ${\displaystyle \ln northward}$ . Withal, this is a result from the contributions of numbers with "abnormally many" divisors.

In abstruse algebra [edit]

Ring theory [edit]

Division lattice [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 past 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 as well [edit]

• Arithmetic functions
• Euclidean algorithm
• Fraction (mathematics)
• Tabular array of divisors — A table of prime number and non-prime number divisors for ane–1000
• Table of prime number factors — A table of prime factors for 1–1000
• Unitary divisor

Notes [edit]

1. ^ ${\displaystyle \gcd }$ refers to the greatest common divisor.

1. ^ ^{a b} Hardy & Wright 1960, p. 1
2. ^ ^{a b} Niven, Zuckerman & Montgomery 1991, p. iv
3. ^ Durbin 2009, p. 57, Affiliate Three Section 10
4. ^ "FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois" (PDF).
5. ^ ${\displaystyle a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b+c=(j+k)a\Rightarrow a\mid (b+c)}$ . Similarly, ${\displaystyle a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b-c=(j-k)a\Rightarrow a\mid (b-c)}$
6. ^ Hardy & Wright 1960, p. 264, Theorem 320

References [edit]

• Durbin, John R. (2009). Modern Algebra: An Introduction (sixth ed.). New York: Wiley. ISBN978-0470-38443-5.
• Richard Yard. Guy, Unsolved Problems in Number Theory (third ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B.
• Hardy, G. H.; Wright, East. K. (1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press.
• Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN0-02-353820-ane
• Niven, Ivan; Zuckerman, Herbert Due south.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons. ISBN0-471-62546-nine.
• Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
• Sims, Charles C. (1984), Abstract Algebra: A Computational Arroyo, New York: John Wiley & Sons, ISBN0-471-09846-nine

What Is A Proper Factor,

Source: https://en.wikipedia.org/wiki/Divisor

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