Definition:Integral Multiple
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Definition
Rings and Fields
Let $\struct {F, +, \times}$ be a ring or a field whose zero is $0_F$.
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:
- $n \cdot a := \begin {cases}
0_F & : n = 0 \\ \paren {\paren {n - 1} \cdot a} + a & : n > 1 \\ \size n \cdot \paren {-a} & : n < 0 \\ \end {cases}$ where $\size n$ 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).