corollary of Bézout’s lemma


If  gcd(a,c)=1  and  c|ab,  then  c|b.

Proof.  Bézout’s lemma ( gives the integers x and y such that  xa+yc=1.  This implies that  xab+ybc=b,  and because here the both summands are divisible by c, so also the sum, i.e. b, is divisible by c .

Note.  A similar theorem holds in all Bézout domains (, also in Bézout rings.

Title corollary of Bézout’s lemma
Canonical name CorollaryOfBezoutsLemma
Date of creation 2013-03-22 14:48:16
Last modified on 2013-03-22 14:48:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Theorem
Classification msc 11A05
Synonym Euclid’s lemma
Synonym product divisible but factor coprime
Related topic GreatestCommonDivisor
Related topic DivisibilityInRings
Related topic DivisibilityByProduct