PlanetMath: corollary of Bézout's lemma

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corollary of Bézout's lemma

(Theorem)

Theorem 1 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.

"corollary of Bézout's lemma" is owned by pahio.

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Other names:

Euclid's lemma, product divisible but factor coprime

Keywords:

divisibility

This object's parent.

Attachments:: divisibility by prime number (Theorem) by pahio

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Cross-references: Bézout rings, theorem, similar, sum, divisible, implies, integers, proof
There are 6 references to this entry.

This is version 12 of corollary of Bézout's lemma, born on 2004-11-09, modified 2008-04-11.
Object id is 6459, canonical name is CorollaryOfBezoutsLemma.
Accessed 3229 times total.

Classification:

AMS MSC:

11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)

Pending Errata and Addenda

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