Definition:Greatest Common Divisor/Integers/Definition 1
Definition
Let $a, b \in \Z: a \ne 0 \lor b \ne 0$.
The greatest common divisor of $a$ and $b$ is defined as:
- the largest $d \in \Z_{>0}$ such that $d \divides a$ and $d \divides b$
where $\divides$ denotes divisibility.
This is denoted $\gcd \set {a, b}$.
When $a = b = 0$, $\gcd \set {a, b}$ is undefined.
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 defined as
Some sources gloss over the fact that at least one of $a$ and $b$ must be non-zero for $\gcd \set{ a, b }$ to be defined.
Some sources insist that both $a$ and $b$ be non-zero or strictly positive.
Some sources define $\gcd \set {a, b} = 0$ for $a = b = 0$.
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.
Also see
- Results about the greatest common divisor can be found here.
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 23$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$: Example $7.8$
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Definition $2 \text{-} 2$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.1$: Algorithms: Algorithm $\text{E}$