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# corollary of Bézout’s lemma

###### Theorem.

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.

Keywords:

divisibility

Related:

GreatestCommonDivisor, DivisibilityInRings, DivisibilityByProduct

Synonym:

Euclid's lemma, product divisible but factor coprime

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

11A05*no label found*

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theorem style by alozano ✓