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单词 GreatestCommonDivisorOfSeveralIntegers
释义

greatest common divisor of several integers


Theorem.  If the greatest common divisorMathworldPlanetmathPlanetmath of the nonzero integers a1,a2,,an is d, then there exist the integers x1,x2,,xn such that

d=x1a1+x2a2++xnan.(1)

Proof.  In the case  n=2  the Euclidean algorithmMathworldPlanetmath for two nonzero integers a,b always guarantees the integers x,y such that

gcd(a,b)=xa+yb.(2)

Make the induction hypothesis that the theorem is true whenever  n<k.
Since obviously

d=gcd(a1,a2,,ak)=gcd(gcd(a1,a2,,ak-1),ak),

we may write by (2) that

d=xgcd(a1,a2,,ak-1)+yak

and thus by the induction hypothesis that

d=x(x1a1+x2a2++xk-1ak-1)+yak.

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).

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更新时间:2025/5/4 14:37:06