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>. | |
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− | == | + | == 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 Also == |
− | + | * [[Arithmetic functions]] | |
− | + | * [[Modular arithmetic]] | |
− | + | * [[Diophantine equations]] | |
− | + | [[Category:Number theory]] | |
− | + | [[Category:Definition]] | |
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Latest revision as of 15:21, 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 .