Modular Arithmetic Definitions and Propositions

1. The ``floor'' function is defined by the formula
   $$\lfloor x \rfloor :=$$   ( the greatest integer less than or equal to $$x).$$
   This is also known as ``the greatest integer function,'' and in old texts is denoted by (whole) brackets. Examples: $\lfloor 3.789 \rfloor = 3$; $\lfloor -3.789 \rfloor = -4$.
   
   
2. The ``mod'' operator is defined as follows:
   $$x \mod y := x - y \cdot \lfloor x/y \rfloor$$
   if $y \ne 0$. For positive integers $x$ and $y$, $x \mod y =$ the remainder in integer division of $x$ by $y$. Examples: $110 \mod 26 = 6$; $-52 \mod 26 = 0$.
   
   
3. The ``mod'' relation is defined as follows:
   $$a \equiv b$$    (mod $$m)$$    if and only if  $$a$$    mod $$m = b$$    mod $$m.$$

   The above definitions make sense even for real numbers. When $a, b, m$ are integers and $m > 0$,

   $$a \equiv b$$    (mod $$m)$$    if and only if  $$a-b$$    is a multiple of  $$m.$$
   Examples: $110 \equiv 6$    (mod $26)$; $-80 \equiv 24$    (mod $26)$.
   
   
4. Let $a, b$ be integers. Then $a \vert b$, read ``$a$ divides $b$,'' if and only if $b$ is a multiple, i.e., an integer multiple, of $a$: $b = ka$ for some integer $k$. Examples: $7 \vert 98$; $-5 \vert 100$; but $8 \nmid 26$ (8 does not divide 26).
   
   
5. Graham, Knuth, and Patashnik's divisibility definition (assume that $a$ and $b$ are integers):
   $$a \backslash b$$    if and only if  $$a > 0$$    and  $$a \vert b.$$

6. The ``greatest common divisor,'' abbreviated gcd, of a set of integers is, of course, the largest positive integer that divides every integer in the set. Examples: $\gcd(24, 52) = 4$; $\gcd(54, 42) = 6$.
   
   
7. Let $a, x, m$ be integers with $m > 0$. Let $g = \gcd(a,m)$. The number of solutions of $a x \equiv b$    (mod $m)$ in the set $\{1, 2, \ldots, m\}$ is 0 if $g \nmid b$ and is $g$ if $g \vert b$.