View source for Mutually-prime numbers

''coprimes, relatively-prime numbers''

Integers without common (prime) divisors. The [[Greatest common divisor|greatest common divisor]] of two coprimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656002.png" /> is 1, which is usually written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656003.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656005.png" /> are coprime, there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m0656009.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065600/m06560010.png" />.

The concept of being coprime may also be applied to polynomials and, more generally, to elements of a [[Euclidean ring|Euclidean ring]].

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR></table>