A Question of Erd\H{o}s and Graham on Covering Systems

Abstract

Erd\H{o}s and Graham (Erd\H{o}s and Graham, 1980) asked if there exists an $n$ such that the divisors of $n$ greater than 1 are the moduli of a distinct covering system with the following property: If there exists an integer which satisfies two congruences in the system, $a\mod d$ and $a'\mod d'$, then $\gcd(d,d')=1$. We show that such an $n$ does not exist. This problem is part of Problem # 204 on the website www.erdosproblems.com, compiled and maintained by Thomas Bloom. We also study when the divisors of $n$ greater than $1$ can form a congruence system satisfying the above condition.