Divisibility

The concept of divisibility founds the Arithmétique.

This article treats divisibility as a whole of the integers ($\backslash \; mathbb\; \{Z\}$).

Definition

Are has and B two entireties. To say that has divides B is equivalent to say that

• has is a divider of B (if has is nonnull)
• B is a multiple of has
• it exists an entirety K such as B = a.k
• $\backslash \; exists\; K\; \backslash \; in\; \backslash \; mathbb\; \{Z\},\; b=ak$
• has | B

Properties of divisibility

Are has , B and C three entireties.

• has | has (Réflexivité)

• - has | has
• has | 0
• 1 | has
• $a|B\; \backslash \; Land\; B|C\; \backslash \; Rightarrow\; has|c$ (Transitivity)
• $a|B\; \backslash \; Rightarrow\; ac|bc$
• $\backslash \; forall\; (U,\; v)\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\; ^2,\; has|B\; \backslash \; Land\; has|C\; \backslash \; Rightarrow\; has|(bu+cv)$ (conservation by linear Combination)
• $a|B\; \backslash \; Land\; B|\backslash \; Rightarrow\; has\; |has|=|B|$ (Antisymétrie)
• the relation of divisibility is a Relation of a partial nature on $\backslash \; mathbb\; \{NR\}$.
• $\backslash \; mathbb\; \{NR\}$ provided with divisibility, the pgcd and PC is a lattice.

See too

• Criteria of divisibility

• Prime numbers
• highest common factor
• Euclidean Division

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