Properties of Division

Theorem (Properties if Division)
Let  
\[a, \; b\]
  be positive integers, and let  
\[c, \; d\]
  be any integers. Then
a) If  
\[a | c\]
  then  
\[a | (c+na)\]
  for any integer  
\[n\]
.
b) If  
\[c \neq 0\]
  and  
\[a | c\]
  then  
\[a \le |c|\]
.
c) If  
\[a | b\]
  and  
\[b | a\]
  then  
\[a=b\]
.
d) If  
\[a | b, \; b | c\]
  then  
\[a | c\]
.
e) If  
\[a | c, \; a | d\]
  then  
\[a | (mc+nd)\]
  for any integers  
\[m, \; n\]
.
Proof
a)  
\[a |c\]
  so there is an integer  
\[q\]
  such that  
\[aq=c\]
. Then  
\[c+na=aq+na=a(q+n)=ax\]
  where  
\[x=qn\]
  is an integer, so  
\[a | (c+na)\]
.
b) Let  
\[c=aq\]
  as in a) then  
\[| c |= | aq |= a | q |\]
  and the result follows since  
\[| q | \ge 1\]
  and  
\[q\]
  is an integer. This result implies that  
\[c\]
  has only a finite number of divisors, since any divisor must be less than or equal to  
\[| c |\]
.
c) If  
\[a | b, \; b | a \]
  then from b)  
\[a \le b, \; b \le a\]
. The result follows.
d) If  
\[a | b, \; b | c \]
  then there are integers  
\[s, \; t\]
  such that  
\[b=as, \; c=bt\]
. Then  
\[c=ast=a(st)\]
  so  
\[a | c\]
.
e) If  
\[a | c, \; a | d \]
  then there are integers  
\[s, \; q\]
  such that  
\[c=as, \; d=aq\]
. Then  
\[mc+nd=mas+naq=a(ms+nq)=ax\]
  where  
\[x=ms+nq\]
  is an integer so e) is proved.