Definition:Coprime

GCD Domain
Let $\left({D, +, \times}\right)$ be a GCD domain.

Let $U \subseteq D$ be the Group of Units of $D$.

Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$

Let $d = \gcd \left\{{a, b}\right\}$ be the greatest common divisor of $a$ and $b$.

Then $a$ and $b$ are coprime, or relatively prime, if $d \in U$.

That is, two elements of a Euclidean domain are coprime iff their greatest common divisor is a unit of $D$.

Integers
Let $a$ and $b$ be integers such that $b \ne 0$ and $a \ne 0$ (i.e. they are not both zero).

Let $\gcd \left\{{a, b}\right\}$ be the greatest common divisor of $a$ and $b$.

If $\gcd \left\{{a, b}\right\} = 1$, then $a$ and $b$ are coprime, or relatively prime.

Alternatively we can say $a$ is prime to $b$, and at the same time that $b$ is prime to $a$.



Relatively Composite
If two integers are not coprime, they are relatively composite.



Notation
If $\gcd \left\{{a, b}\right\} = 1$, then the notation $a \perp b$ is encouraged.

If $\gcd \left\{{a, b}\right\} \ne 1$, the notation $a \not \perp b$ can be used.

Coprime as a Relation
It can be seen that considered as a relation, $\perp$ is:


 * $(1): \quad$ Non-reflexive: $a \not \perp a$ except when $a = \pm 1$
 * $(2): \quad$ Symmetric: $a \perp b \iff b \perp a$
 * $(3): \quad$ Not antisymmetric: $a \perp b \land b \perp a \not \implies a = b$
 * $(4): \quad$ Non-transitive: Consider $2 \perp 3, 3 \perp 4, 2 \not \perp 4$ and $2 \perp 3, 3 \perp 5, 2 \perp 5$.

Definition on Euclidean Domain

 * : $\S 6.29$

Definition on Integers

 * : $\S 1.1$: Example $4$
 * : $\S 1.2.4$
 * : $\S 3.12$
 * : $\S 2.2$: Definition $2.3$
 * : $\S 23$
 * : $\S 12$