Erdős Problem #204

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 \((d,d')=1\). The density of such \(n\) is zero. Erdős and Graham believed that no such \(n\) exist. Adenwalla [Ad25] has proved there are no such \(n\). This was formalized by van Doorn in Lean using Aristotle.