corollary of Bézout’s lemma

Theorem.

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

Proof.  Bézout’s lemma (http://planetmath.org/BezoutsLemma) 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 (http://planetmath.org/BezoutDomain), 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