Divide && Multiple
Suppose
\(
m,n\in Z, n\neq 0,
\)
\(
\forall k (m=nk)
\)
\(
n \text{ divide } m
\)
Sign as \( n\mid m \)
n is factor from m.
m is multiple from n.
Otherwise, \( n\nmid m \)
Suppose \( m,n,r\in Z \)
-
\(
1\mid m
\)
\(
m\mid 0
\)
-
If
\(
((m\mid n)
\text{ and }
(n\mid m))
\rightarrow
(m= \pm n)
\)
-
\(
(
(m\mid n)
\text{ and }
(n\mid r))
\rightarrow
m\mid r
\)
-
\(
(m\mid n)
\rightarrow
(m\mid nx,\forall x\in Z)
\)
-
\(
((m\mid n)\text{ and }(m\mid r))
\rightarrow
(m\mid (nx+ry),\forall x,y\in Z)
\)
-
\(
\because m\mid n \text{ and }
m\mid r
\)
\(
\therefore \exists p,q\in Z
\rightarrow n=pm \text{ and } r=qm
\)
\(
\rightarrow
n+r = pm+qm
\)
\(
\rightarrow
nx+ry = pmx+qmy
=
m(px+qy)
\)
\(
\therefore m\mid nx+ry
\)