# Definition:Integral Multiple

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## Definition

### Rings and Fields

Let $\left({F, +, \times}\right)$ be a ring or a field.

Let $a \in F$.

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

Then $n \cdot a$ is an integral multiple of $a$ where $n \cdot a$ is defined as in Powers of Ring Elements:

$n \cdot a := \begin{cases} 0_F & : n = 0 \\ \left({\left({n - 1}\right) \cdot a}\right) + a & : n > 1 \\ \left|{n}\right| \cdot \left({-a}\right) & : n < 0 \\ \end{cases}$

where $\left|{n}\right|$ is the absolute value of $n$.

### Real Numbers

This concept is often seen when $F$ is the set of real numbers $\R$.

Let $x, y \in \R$ be real numbers.

Then $x$ is an integral multiple of $y$ if and only if $x$ is congruent to $0$ modulo $y$:

$x \equiv 0 \pmod y$

That is:

$\exists k \in \Z: x = 0 + k y$

## Also see

Compare divisor, in which all the numbers involved are integers (or at least, elements of an integral domain).