Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'?

That is, for all $d\mid n$ with $d>1$ there is an associated $a_d$ such that every integer is congruent to some $a_d\pmod{d}$, and if there is some integer $x$ with

$$x\equiv a_d\pmod{d}\textrm{ and }x\equiv a_{d'}\pmod{d'}$$

then $(d,d')=1$.