Definition:Modulo Division/Divisor

Definition

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \set {0, 1, \ldots, m - 1}$

For all $a, b \in \Z_m$, let $a \div_m b$ denote the operation of division modulo $m$:

$a \div_m b$ equals the integer $q \in \Z_m$ such that $b \times_m q \equiv a \pmod m$

The integers $b$ and $q$ are divisors of $a$ modulo $m$.

Examples

5 modulo 12

In modulo $12$ division, $5$ has the following divisors:

$1, 5, 7, 11$

8 modulo 12

In modulo $12$ division, $8$ has the following divisors:

$1, 2, 4, 5, 7, 8, 10, 11$

Also known as

A divisor modulo $m$ is also known as a factor modulo $m$.

Also see

• Results about divisors modulo $m$ can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor modulo $n$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor modulo $n$