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Let $a,b$ and $c$ be positive integers such that $gcd(a,b)=1$ and $a\vert bc$. Prove that $a \vert c$.

Bill Dubuque
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Kungu jackson
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    Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – AugSB Jun 01 '17 at 07:28
  • Look at the prime divisors of a,b,c – wonko Jun 01 '17 at 07:28
  • this is Euclid's Lemma https://en.wikipedia.org/wiki/Euclid%27s_lemma – Marcos TV Jun 01 '17 at 07:29
  • If $ua+vb=1$ then $c=uac+vbc=(uc+v\frac{bc}a)\cdot a$ – Hagen von Eitzen Jun 01 '17 at 07:31
  • It's Gauß'lemma. Euclid'slemma is the case $a$ prime. – Bernard Jun 01 '17 at 07:33

1 Answers1

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As $a|bc$, we have

$\gcd(a,bc)=a$

and

$\gcd(a,bc)\le\gcd(a,b)\gcd(a,c)=\gcd(a,c)$

Because $a\le c$, $\gcd(a,c)\le a$

Therefore $\gcd(a,c)=a$, therefore $a|c$.

JMP
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