Definition:Common Divisor/Integers

Definition

Let $S$ be a finite set of integers, that is:

$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \Z}$

Let $c \in \Z$ such that $c$ divides all the elements of $S$, that is:

$\forall x \in S: c \divides x$

Then $c$ is a common divisor of all the elements in $S$.

Also known as

A common divisor is also known as a common factor.

In Euclid's The Elements, the term common measure is universally used for this concept.

Examples

$20$, $70$ and $80$

The integers $20$, $70$ and $80$ have $2$, $5$ and $10$ as common divisors.

Also see

• Results about common divisors can be found here.

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Example $\text {2-4}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 12$: Highest common factors and Euclid's algorithm
• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): common factor (common divisor)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): common factor (common divisor)