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Difference between revisions of "Divisibility"

(I want to overhaul this stub into a ginormous handout on divisibility.)
 
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== Definition ==
 
== 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>.
 
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>.
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=== Examples ===
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* <math>6</math> divides <math>48</math> as <math>6 \times 8 = 48</math>, so we may write that <math>6 \mid 48</math>.
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* <math>-2</math> divides <math>6</math> as <math>6/(-2) = -3</math>, so we may write that <math>-2 \mid 6</math>.
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* The positive divisors of <math>35</math> are <math>1</math>, <math>5</math>, <math>7</math>, and <math>35</math>.
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* By convention, we write that every nonzero integer divides <math>0</math>; so <math>-1923 \mid 0</math>.
  
 
== See Also ==
 
== See Also ==
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* [[Diophantine equations]]
 
* [[Diophantine equations]]
  
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[[Category:Number Theory]]
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[[Category:Number theory]]
 
[[Category:Definition]]
 
[[Category:Definition]]

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 $a$ is an integer $b$ that can be multiplied by some integer to produce $a$. We may equivalently state that $a$ is a multiple of $b$, and that $a$ is divisible or evenly divisible by $b$.

Definition

An integer $a$ is divisible by a nonzero integer $b$ if there exists some integer $n$ such that $a = bn$. We may write this relation as \[b \mid a.\] An alternative definition of divisibility is that the fraction $a / b$ is an integer — or using modular arithmetic, that $b \equiv 0 \pmod a$. If $b$ does not divide $a$, we write that $b \nmid a$.

Examples

  • $6$ divides $48$ as $6 \times 8 = 48$, so we may write that $6 \mid 48$.
  • $-2$ divides $6$ as $6/(-2) = -3$, so we may write that $-2 \mid 6$.
  • The positive divisors of $35$ are $1$, $5$, $7$, and $35$.
  • By convention, we write that every nonzero integer divides $0$; so $-1923 \mid 0$.

See Also

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