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Difference between revisions of "Divisibility"

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'''Divisibility''' is the ability of a number to be evenly divided by another number. For example, four divided by two is equal to two, an [[integer]], and therefore we say four ''is divisible by'' two.
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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,
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the study of [[arithmetic functions]], [[modular arithmetic]], and [[Diophantine equations]] all depend on divisibility for rigorous foundation.
  
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==Divisibility Videos==
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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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https://youtu.be/bIipw2XSMgU
 
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https://youtu.be/6xNkyDgIhEE?t=1699
 
  
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== Notation ==
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== Definition ==
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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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We commonly write <math>n|k</math>. This means that <math>n</math> is a [[divisor]] of <math>k</math>. So for the example above, we would write 2|4.
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== See Also ==
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* [[Arithmetic functions]]
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* [[Modular arithmetic]]
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* [[Diophantine equations]]
  
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== See also ==
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[[Category:Number Theory]]
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* [[Divisor]]
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[[Category:Definition]]
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* [[Divisibility rules]]
 
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* [[Number theory]]
 

Revision as of 15:06, 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 $b \nmid a$.

See Also

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