cohen-macaulayness of reduced and non-reduced schemes Let $X$ be a Cohen-Macaulay scheme (let's say of finite type over a field).
Let $X_{red}$ be the corresponding reduced scheme. Is it true that $X_{red}$ is also 
Cohen-Macaulay? 
 A: The simplest counter-example I know is the following: Hartshorne showed that if $k$ has positive characteristic, $k[s^4, s^3t, st^3,t^4]$ (which will be $X_{red}$) is a set-theoretic complete intersection (said complete intersection will be $X$). The former is well-known to be not CM (cheapest proof: $s^4,t^4$ form a s.o.p but not a regular sequence). 
There are more examples of projective curves which are set-theoretic c.i. (you can  find quite a few papers). Among them the ones which are not arithmetically CM give counter examples via taking the affine cone.  
EDIT (to address the OP's new question below): this new situation is discussed in my answer quoted above by Vesselin, so you may want to take a look. To use that answer's notation, you need at least two height one primes, say  $P,Q$, which are Cohen-Macaulay, and $a[P]+b[Q]=0$ in the class group of $Y$ (this takes care of the assumption that $X$ is set theoretically principal), but $[P]+[Q]$ is not CM. Such examples probably still exist (for example there are torsion classes which are not CM), but we may need a lot of luck (or hard work) to write one down.    
