Definition:Coprime/Integers

Definition
Let $a$ and $b$ be integers such that $b \ne 0$ and $a \ne 0$ (i.e. they are both non-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.



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 $a$.

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 $a \not \!\! \mathop{\perp} b$ can be used.

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


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