Editing Coprime integers (section)

== Coprimality in sets ==
<!-- [[Coprime set]] redirects here -->

A [[Set (mathematics)|set]] of integers <math>S=\{a_1,a_2, \dots, a_n\}</math> can also be called ''coprime'' or ''setwise coprime'' if the [[greatest common divisor]] of all the elements of the set is 1. For example, the integers 6, 10, 15 are coprime because 1 is the only positive integer that divides all of them.

If every pair in a set of integers is coprime, then the set is said to be ''pairwise coprime'' (or ''pairwise relatively prime'', ''mutually coprime'' or ''mutually relatively prime''). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true. For example, the integers 4, 5, 6 are (setwise) coprime (because the only positive integer dividing ''all'' of them is 1), but they are not ''pairwise'' coprime (because {{math|1=gcd(4, 6) = 2}}).

The concept of pairwise coprimality is important as a hypothesis in many results in number theory, such as the [[Chinese remainder theorem]].

It is possible for an [[infinite set]] of integers to be pairwise coprime. Notable examples include the set of all prime numbers, the set of elements in [[Sylvester's sequence]], and the set of all [[Fermat numbers]].