Definition 1.8.1.53.  Let m, n ∈ . We define:
m and n are coprime  :⇔  ∀ a ∈ + s.t. a ∣ m and a ∣ n : a = 1  ⇔  ∄ b ∈ >1 : [b ∣ m and b ∣ n]  ⇔  Div(m) ∩ Div(n) ⊆ {1}  ⇔  Div(m) ∩ Div(n) = {1}  ⇔  ∄ primep : [p ∣ m and p ∣ n]
Equivalence.  No proof.
Remarks.

Note that in addition to this definition for natural numbers, there is a compatible definition for integers: $../Integer/coprime.

References.