Definition:Divisor/Integers

Definition

Let $\left({\Z, +, \times}\right)$ be the integral domain of integers.

Let $x, y \in \Z$.

Then $x$ divides $y$ is defined as:

$x \mathop \backslash y \iff \exists t \in \Z: y = t \times x$

The conventional notation for this is "$x \mid y$", but there is a growing trend to follow the notation above, as espoused by Knuth etc.

If $x \mathop \backslash y$, then:

• $x$ is a divisor (or factor) of $y$
• $y$ is a multiple of $x$
• $y$ is divisible by $x$.

To indicate that $x$ does not divide $y$, we write $x \nmid y$.

References

"The notation '$m \mid n$' is actually much more common than '$m \mathop \backslash n$' in current mathematics literature. But vertical lines are overused - for absolute values, set delimiters, conditional probabilities, etc. - and backward slashes are underused. Moreover, '$m \mathop \backslash n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward."

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 24$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.1$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 3.10$
• George E. Andrews: Number Theory (1971): $\S 2.2$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 22$
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.3$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 11$