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Def/Divides
From SlugmathWiki
Definition of Divides: If $x$ and $y$ are integers, we say that $x$ divides $y$ if there exists an integer $m$ such that $y = mx$.
More generally, one often uses the word "divides" in the context of any commutative ring $R$. In this generality, the sentence "$x$ divides $y$", for two elements $x,y \in R$, means that: $$\exists m \in R, \mbox{ such that } y = xm.$$
The following are equivalent ways of stating that $x$ divides $y$:
Logical Connections
This definition logically relies on the following definitions and statements: Def/Integer
The following statements and definitions logically rely on the material of this page: Def/Even, Def/Greatest common divisor, Def/Irreducible element, Def/Prime number, Def/Relatively prime, State/Classification of subgroups of the integers, State/Counting Sylow subgroups, State/Divisibility corresponds to containment of principal ideals, State/Lagranges theorem, State/Multiplication increases size, State/Mutual divisibility of natural numbers implies equality, and State/Two out of three principle for divisibility
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic number theory, Clust/Basic ring theory
