greatest common divisor of several integers
Proof. In the case the Euclidean algorithm for two nonzero integers always guarantees the integers such that
Make the induction hypothesis that the theorem is true whenever .
we may write by (2) that
and thus by the induction hypothesis that
Consequently, we have gotten an equation of the form (1), Q.E.D.
Note. An analogous theorem concerns elements of any Bézout domain (http://planetmath.org/BezoutDomain).
|Title||greatest common divisor of several integers|
|Date of creation||2013-03-22 19:20:14|
|Last modified on||2013-03-22 19:20:14|
|Last modified by||pahio (2872)|
|Synonym||GCD of several integers|