greatest common divisor of several integers
Theorem. If the greatest common divisor of the nonzero integers is , then there exist the integers such that
(1) |
Proof. In the case the Euclidean algorithm for two nonzero integers always guarantees the integers such that
(2) |
Make the induction hypothesis that the theorem is true whenever .
Since obviously
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).