Definition:Greatest Common Divisor/Integers

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Definition

Let $a, b \in \Z: a \ne 0 \lor b \ne 0$.


Definition 1

The greatest common divisor of $a$ and $b$ is defined as:

the largest $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$


Definition 2

The greatest common divisor of $a$ and $b$ is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that:

$(1): \quad d \divides a \land d \divides b$
$(2): \quad c \divides a \land c \divides b \implies c \divides d$


This is denoted $\gcd \set {a, b}$.


When $a = b = 0$, $\gcd \set {a, b}$ is undefined.


In the above, $\divides$ denotes divisibility.


General Definition

This definition can be extended to any (finite) number of integers.

Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\exists x \in S: x \ne 0$ (that is, at least one element of $S$ is non-zero).

Definition 1

The greatest common divisor of $S$:

$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$

is defined as the largest $d \in \Z_{>0}$ such that:

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

where $\divides$ denotes divisibility.


Definition 2

The greatest common divisor of $S$:

$\gcd \paren S = \gcd \set {a_1, a_2, \ldots, a_n}$

is defined as the (strictly) positive integer $d \in \Z_{>0}$ such that:

  \(\ds \forall x \in S:\) \(\ds d \divides x \)      
  \(\ds \forall e \in \Z: \forall x \in S:\) \(\ds e \divides x \implies e \divides d \)      

where $\divides$ denotes divisibility.


By convention:

$\map \gcd \O = 1$


Also known as

The greatest common divisor is often seen abbreviated as GCD, gcd or g.c.d.

Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs.

The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable.


The greatest common divisor is also known as the highest common factor, or greatest common factor.

Highest common factor when it occurs, is usually abbreviated as HCF, hcf or h.c.f.

It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$.


The archaic term greatest common measure can also be found, mainly in such as Euclid's The Elements.


Examples

$12$ and $-8$

The greatest common divisor of $12$ and $-8$ is:

$\gcd \set {12, -8} = 4$


$-12$ and $30$

The greatest common divisor of $-12$ and $30$ is:

$\gcd \set {-12, 30} = 6$


$-5$ and $5$

The greatest common divisor of $-5$ and $5$ is:

$\gcd \set {-5, 5} = 5$


$8$ and $17$

The greatest common divisor of $8$ and $17$ is:

$\gcd \set {8, 17} = 1$

That is, $8$ and $17$ are coprime.


$-8$ and $-36$

The greatest common divisor of $-8$ and $-36$ is:

$\gcd \set {-8, -36} = 4$


$n$ and $0$

Let $n \in \Z_{>0}$.

The greatest common divisor of $n$ and $0$ is:

$\gcd \set {n, 0} = n$


Also see

Thus it can be seen that for any two GCDs $d$ and $d'$ we have that $d = \pm d'$.

  • Results about the greatest common divisor can be found here.


Sources

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