The Mirsky-Newman theorem states that a finite exact covering system of the integers by residue classes with moduli at least 2 must contain two classes with the same largest modulus (Newman 1971). In particular, there is no exact covering system with all moduli distinct. Here an exact covering system means that each integer belongs to exactly one class.
To see the obstruction, normalize the classes as 
with 
. The generating
function identity for the nonnegative integers,
valid for 
,
is
 |
If the largest modulus 
occurred only once, multiplying by 
and letting 
tend to a primitive Nth
root of unity would give a nonzero value on the left and zero on the right. All
smaller moduli have denominators nonzero at this root
of unity.
This is problem 947 in the Erdős problems collection (Bloom 2026).
