Definition:Integer Division
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Definition
Let $a, b \in \Z$ be integers such that $b \ne 0$..
From the Division Theorem:
- $\exists_1 q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$
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 $-7$
\(\ds 1 \div \paren {-7}\) | \(=\) | \(\ds 0 \rem 1\) | ||||||||||||
\(\ds -2 \div \paren {-7}\) | \(=\) | \(\ds 1 \rem 5\) | ||||||||||||
\(\ds 61 \div \paren {-7}\) | \(=\) | \(\ds -8 \rem 5\) | ||||||||||||
\(\ds -59 \div \paren {-7}\) | \(=\) | \(\ds 9 \rem 4\) |