Integer Divisors

Definition

Suppose $a$ is an integer. A divisor of $a$ is another integer $b$ for which there exists an integer $k$ satisfying

\[ a = kb \]

In this case, we say $b$ divides $a$ or $a$ is divisible by $b$. Often, we specialize to the case in which all the integers involved are positive, but the definition we give here also allows for some to be negative.

Suppose that $c \in \Z $ does not divide $a$. Denote by $k$ the largest integer satisfying

\[ kc < a \]

Define $r = a - kc$. We call $r > 0$ the remainder of dividing $a$ into $b$.

Notation

For $a, b \in \Z $ we denote that $b$ divides $a$ by writing $b \mid a$. For $c \in \Z $, we denote that $c$ does not divide $a$ by $c \not\mid a$. We denote the remainder of dividing $c$ into $a$ by $a \mod c$.