Difference between revisions of "Divisibility"

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 $-1923 \mid 0$ .

@@ Line 1: / Line 1: @@
-'''Divisibility''' is the ability of a number to be evenly divided by another number.  For example, four divided by two is equal to two, and therefore, four is divisible by two.
+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.
-== Notation ==
+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>.
-We commonly write <math>n|k</math>. This means that n is a divisor of k. So for the example above, we would write 2|4.
+== 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 ==
+== See Also ==
-* [[Divisibility rules]]
+* [[Arithmetic functions]]
-* [[Number theory]]
+* [[Modular arithmetic]]
+* [[Diophantine equations]]
+[[Category:Number theory]]
+[[Category:Definition]]

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Difference between revisions of "Divisibility"

Latest revision as of 15:21, 29 April 2023

Definition

Examples

See Also