# Definition:Integer Division

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## Definition

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

From the Division Theorem:

- $\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$

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

The process of finding $q$ and $r$ is known as **(integer) division of $a$ by $b$**, and we write:

- $a \div b = q \rem r$

### Exact Division

Let:

- $a \div b = q \rem 0$

where $\div$ denotes **integer division**.

This is an instance of **exact division**.

## Notation

The operation of division can be denoted as:

- $a / b$, which is probably the most common in the general informal context

- $\dfrac a b$, which is the preferred style on $\mathsf{Pr} \infty \mathsf{fWiki}$

- $a : b$, which is usually used when discussing ratios

- $a \div b$, which is rarely seen outside grade school, but can be useful in contexts where it is important to be specific.

## Examples

### $29$ Divided by $8$

- $29 \div 8 = 3 \rem 5$

### Division by $-7$

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

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

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

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

## Also see

- Definition:Division over Euclidean Domain of which this is an example, with the Euclidean valuation being the absolute value

- Results about
**integer division**can be found**here**.

## Linguistic Note

The verb form of the word **division** is **divide**.

Thus to **divide** is to perform an act of **division**.