# Coprimality Two integers are **coprime** if they have no common factors (beside 1), or, equivalently, if their *gcd* is 1. A set of integers, e.g. `A = {3, -7, 1}`, is *relatively prime* if there is no integer `d` > 1 (i.e. `d ∈ ℕᐩᐩ`), such that `d` divides all of them. Relatively Prime is also called "coprime" or "mutually prime". ¬∃x ∈ ℕᐩᐩ. ∀a ∈ A ⊆ ℤ. x ∤ a ...then `a`'s are relatively prime A set of integers is *coprime* if no integer `d` > 1 divides any pair of them. Therefore, for any two integers these two definitions coincide, but for a larger collection, the property of coprimality is stronger because a set of integers can be relatively prime but not coprime, e.g. (6, 15, -5). The *coprime* relation is a relation between (two) integers. It can also be considered a constraint on a set of integers. ## Linguistics * **coprimeness** or **coprimality** Property/quality/condition of being coprime * The word *"coprime"*, alternatively spelled *"co-prime"* is a non-comparable adjective. In math, adjectives can be used in phrases like "A and B are coprime", "A is coprime to B", or less commonly, "A is coprime with B". * (number theory) *"coprime"* may be used to *relate* two or more natural numbers as having no natural number factors in common (besides 1), as in "24 and 35 are coprime". * (number theory) *"coprime"* may be used to qualify a natural number having no natural number factors (except for 1) in common with some other natural numbers, as in "10 is coprime to 19". * (algebra) by extension from number theory, one can speak of two or more polynomials whose gcd is a nonzero constant (a polynomial of degree 0).