Definitions

• ${\bf N}=\{1,2,3,\ldots\}$ and is called the set of natural numbers.
• ${\bf Z}=\{\ldots ,-3,-2,-1,0,1,2,3,\ldots\}$ and is called the set of integers.
• $a<b$ means that $b-a\in{\bf N}$.
• $a>b$ means that $a-b\in{\bf N}$.
• $a\equiv b\pmod{m}$ means that a=b+mq for some $q\in{\bf Z}$.
• $a\vert b$ means that there is an integer $q$ such that $b=aq$.
• ${ord}_a (b) = c$ means that $c$ has the following properties:
  • $b^c\equiv 1\pmod{a}$.
  • $c$ is the least such positive integer.
  Note that ${ord}_a (b)$ exists if and only if (iff) $(a,b)=1$.
• $(a,b)=d$ means that $d$ has the following properties:
  • $d\vert a$.
  • $d\vert b$.
  • If $c\vert a$ and $c\vert b$ and $c\in{\bf N}$ then $c\leq d$.
  $d$ is called the greatest common divisor(GCD) of $a$ and $b$.
• $[a,b]=e$ means that $e$ has the following properties:
  • $e>0$.
  • $a\vert e$.
  • $b\vert e$.
  • If $a\vert c$ and $b\vert c$ and $c\in{\bf N}$ then $e\leq c$.
  $e$ is called the least common multiple(LCM) of $a$ and $b$.