Proposition: Divisibility Laws

The following laws of divisibility hold in \(\mathbb Z\):

  1. \((1\mid n\wedge n\mid n\wedge n\mid 0)\quad\forall n\in\mathbb Z\).
  2. \((m\mid n \wedge n\mid m) \Rightarrow n=m\).
  3. \((m\mid n \wedge n\mid l) \Rightarrow m\mid l\).
  4. \(m\mid n\ \Rightarrow (m\mid (nr)\quad\forall r\in\mathbb Z)\).
  5. \((m\mid n \wedge m\mid l) \Rightarrow m\mid (n+l)\).
  6. \((m\mid n \wedge m\mid (n+l)) \Rightarrow m\mid l\).
  7. \(mr\mid nr \Rightarrow m\mid n\).
  8. \((m\mid n\wedge r\neq 0) \Rightarrow mr\mid nr\).
  9. \((m\mid n\wedge r\neq 0) \Rightarrow mr\mid nr\).

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Propositions: 15 16 17 18


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927