Title: On covering systems of integers,
Abstract: A covering system of the integers is a finite collection of modular residue
classes $\{a_m \bmod{m}\}_{m \in S}$ whose union is all integers. Given a
finite set $S$ of moduli, it is often difficult to tell whether there is a
choice of residues modulo elements of $S$ covering the integers. Hough has
shown that if the smallest modulus in $S$ is at least $10^{16}$, then there is
none. However, the question of whether there is a covering of the integers with
all odd moduli remains open. We consider multiplicative restrictions on the set
of moduli to generalize Hough's negative solution to the minimum modulus
problem. In particular, we find that every covering system of the integers has
a modulus divisible by a prime number less than or equal to $19$. Hough and
Nielsen have shown that every covering system has a modulus divisible by either
$2$ or $3$.