Definition:Common Divisor/Integers

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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.


$20$, $70$ 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.