Let $k$ be an algebraically closed field of characteristic zero. Let $a,b,c$ be relatively prime positive integers. Then, is $\dfrac{k[x,y,z]}{(x^a +y^b+z^c)} $ a UFD?
This question is motivated from $(K[x,y,z]/(x^2+y^3+z^7))_{(x,y,z)}$ is a UFD., however I do not know how to make a general argument work.
Further thoughts: I can easily see that the ring is a normal domain (it is Cohen-Macaulay and also regular in co-dimension $1$ by calculating the Jacobian). So I would be done if I can show the divisor class group is $0$.
Please help.