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单词 AlternativeProofOfEuclidsLemma
释义

alternative proof of Euclid’s lemma


We give an alternative proof (see Euclid’s lemma proof), which does not use the Fundamental Theorem of ArithmeticMathworldPlanetmath (since, usually, Euclid’s lemma is used to prove FTA).

Lemma 1.

If abc and gcd(a,b)=1 then ac.

Proof.

By assumptionPlanetmathPlanetmath gcd(a,b)=1, thus we can use Bezout’s lemma to find integers x,y such that ax+by=1. Hence c(ax+by)=c and acx+bcy=c. Since aa and abc (by hypothesis), we conclude that aacx+bcy=c as claimed.∎

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