- Divisibility
- Let \(a, b \in \mathbb{Z} \) \(b\vert a\) iff \( \exists k \in\mathbb{Z} \ni a=bk \).
- I. \( ac \vert b \Rightarrow a \vert b \) and \( ac \vert b \Rightarrow c \vert b \)
- Proof: By definition \( ac \vert b \) iff \( \exists n \in \mathbb{Z} \ni b=acn \). If \( cn=k \in \mathbb{Z} \) then \( b=acn=ak \), so \( a \vert b \)
Proof: By definition \( ac \vert b \) iff \( \exists n \in \mathbb{Z} \ni b=acn \). If \( an=k \in \mathbb{Z} \) then \( b=acn=ck \), so \( c \vert b \)
- II. If \( a \vert b \) and \( a \vert c \) then \( a \vert \left( b + c \right) \)
- Proof: By definition \( a \vert b \) iff \( \exists k \in \mathbb{Z} \ni b=ak \) and \( a \vert c \) iff \( \exists n \in \mathbb{Z} \ni c=an \). Since \( ak + an = b + c = a \left( k + n \right) \) and \( \left( k+ n \right)\in\mathbb{Z} \) then by definition \( a \vert \left(b+c \right)\).
- III. If \( a \vert b \) and \( a \vert c \) then \( a \vert \left( b – c \right) \)
- Proof: By definition \( a \vert b \) iff \( \exists k \in \mathbb{Z} \ni b=ak \) and \( a \vert c \) iff \( \exists n \in \mathbb{Z} \ni c=an \). Since \( ak – an = b – c = a \left( k – n \right) \) and \( \left( k – n \right)\in\mathbb{Z} \) then by definition \( a \vert \left(b-c \right)\).
- IV. If \( a \vert b \) and \( a \vert c \) then \( a \vert bc \)
- Proof: By definition \( a \vert b \) iff \( \exists k \in \mathbb{Z} \ni b=ak \) and \( a \vert c \) iff \( \exists n \in \mathbb{Z} \ni c=an \). Since \( bc =ak \cdot an =at \) and \( kan=t\in\mathbb{Z} \) then by definition \( a \vert bc\).
- V. \( a \vert b \Rightarrow a \vert bc \)
- Proof: By definition \( a \vert b \) iff \( \exists k \in \mathbb{Z} \ni b=ak \). Since \( bc=akc=am \) and \( kc=m\in\mathbb{Z} \) then by definition \( a \vert bc\).
- VI. If \( a \vert b \) and \( a \vert c \) then \( a^2 \vert bc \)
- Proof: By definition \( a \vert b \) iff \( \exists k \in \mathbb{Z} \ni b=ak \) and \( a \vert c \) iff \( \exists n \in \mathbb{Z} \ni c=an \). Since \( bc=ak \cdot an=a^2t\) where \( kn=t\in\mathbb{Z} \) then by definition \( a^2 \vert bc\).