Difference between revisions of "Divisibility"
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− | ''' | + | In [[number theory]], '''divisibility''' is the ability of a number to evenly divide another number. The study of divisibility resides at the heart of number theory, constituting the backbone to countless fields of mathematics. Within number theory, |
+ | the study of [[arithmetic functions]], [[modular arithmetic]], and [[Diophantine equations]] all depend on divisibility for rigorous foundation. | ||
− | + | A '''divisor''' of an integer <math>a</math> is an integer <math>b</math> that can be multiplied by some integer to produce <math>a</math>. We may equivalently state that <math>a</math> is a '''multiple''' of <math>b</math>, and that <math>a</math> is '''divisible''' or '''evenly divisible''' by <math>b</math>. | |
− | + | == Definition == | |
+ | An integer <math>a</math> is divisible by a nonzero integer <math>b</math> if there exists some integer <math>n</math> such that <math>a = bn</math>. We may write this relation as <cmath>b \mid a.</cmath> An alternative definition of divisibility is that the fraction <math>a / b</math> is an integer — or using [[modular arithmetic]], that <math>b \equiv 0 \pmod a</math>. If <math>b</math> does ''not'' divide <math>a</math>, we write that <math>b \nmid a</math>. | ||
+ | === Examples === | ||
+ | * <math>6</math> divides <math>48</math> as <math>6 \times 8 = 48</math>, so we may write that <math>6 \mid 48</math>. | ||
+ | * <math>-2</math> divides <math>6</math> as <math>6/(-2) = -3</math>, so we may write that <math>-2 \mid 6</math>. | ||
+ | * The positive divisors of <math>35</math> are <math>1</math>, <math>5</math>, <math>7</math>, and <math>35</math>. | ||
+ | * By convention, we write that every nonzero integer divides <math>0</math>; so <math>-1923 \mid 0</math>. | ||
− | == See | + | == See Also == |
− | * [[ | + | * [[Arithmetic functions]] |
− | * [[ | + | * [[Modular arithmetic]] |
− | * [[Number theory]] | + | * [[Diophantine equations]] |
+ | |||
+ | [[Category:Number theory]] | ||
+ | [[Category:Definition]] |
Latest revision as of 16:21, 29 April 2023
In number theory, divisibility is the ability of a number to evenly divide another number. The study of divisibility resides at the heart of number theory, constituting the backbone to countless fields of mathematics. Within number theory, the study of arithmetic functions, modular arithmetic, and Diophantine equations all depend on divisibility for rigorous foundation.
A divisor of an integer is an integer
that can be multiplied by some integer to produce
. We may equivalently state that
is a multiple of
, and that
is divisible or evenly divisible by
.
Definition
An integer is divisible by a nonzero integer
if there exists some integer
such that
. We may write this relation as
An alternative definition of divisibility is that the fraction
is an integer — or using modular arithmetic, that
. If
does not divide
, we write that
.
Examples
divides
as
, so we may write that
.
divides
as
, so we may write that
.
- The positive divisors of
are
,
,
, and
.
- By convention, we write that every nonzero integer divides
; so
.