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We're in an integral domain with unity 1 $\neq$ 0. Suppose that the highest common factor between x,y is 1 and the highest common factor for x,z is 1.

Show that $x \mid yz$ implies that $x$ is a unit, or provide a counterexample.

I'm stuck. I don't have that we are in a unique factorization domain, I don't have that this ring is Noetherian.

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    @Henning In any ring, $\rm\:gcd(a,b) = 1\:$ means $\rm\:c\:|\:a,b\:\Rightarrow\:c\:|\:1.\:$ This follows from the [universal definition](http://math.stackexchange.com/a/88411/242) of the gcd.2012-04-14

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Hint $\ $ For the special case $\rm\:x\:$ is irreducible, a counterexample would be an irreducible element that is not prime. These are easy to find in non-UFD number rings. See also this post.

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    This is interesting. So then I am clearly stuck on$a$problem. I am asked to find that if $a$ and $a$ have a highest common factor, then they have a least common multiple. So here is my strategy: I let $(a,b)$ be the hcf, write $(a,b)a'=a$ and $(a,b)b'=b$ and claim that my candidate is $(a,b)a'b'$ is my lcm. I will repost this as a new question.2012-04-14