Definition:Coprime/Integers

Definition
Let $a$ and $b$ be integers such that $b \ne 0$ and $a \ne 0$ (that is, they are both non-zero).

Let $\gcd \set {a, b}$ denote the greatest common divisor of $a$ and $b$.

Then $a$ and $b$ are coprime $\gcd \set {a, b} = 1$.



Also known as
The statement $a$ and $b$ are coprime can also be expressed as:


 * $a$ and $b$ are relatively prime
 * $a$ is prime to $b$, and at the same time that $b$ is prime to $a$.

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


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