# Definition:Integer Division

Jump to navigation
Jump to search

## 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 $-7$

\(\displaystyle 1 \div \paren {-7}\) | \(=\) | \(\displaystyle 0 \rem 1\) | |||||||||||

\(\displaystyle -2 \div \paren {-7}\) | \(=\) | \(\displaystyle 1 \rem 5\) | |||||||||||

\(\displaystyle 61 \div \paren {-7}\) | \(=\) | \(\displaystyle -8 \rem 5\) | |||||||||||

\(\displaystyle -59 \div \paren {-7}\) | \(=\) | \(\displaystyle 9 \rem 4\) |