Definition:Integer Division

Definition

Let $a, b \in \Z$ be integers such that $b \ne 0$..

From the Division Theorem:

$\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$

where $q$ is the quotient and $r$ is the remainder.

The process of finding $q$ and $r$ is known as (integer) division.

Examples

$29$ Divided by $8$

$29 \div 8 = 3 \rem 5$

Division by $-7$

\(\ds 1 \div \paren {-7}\)

\(=\)

\(\ds 0\)

\(\ds \rem 1\)

\(\ds -2 \div \paren {-7}\)

\(=\)

\(\ds 1\)

\(\ds \rem 5\)

\(\ds 61 \div \paren {-7}\)

\(=\)

\(\ds -8\)

\(\ds \rem 5\)

\(\ds -59 \div \paren {-7}\)

\(=\)

\(\ds 9\)

\(\ds \rem 4\)