Erdős Problem #947

There is no exact covering system - that is, a finite collection of congruence classes \(a_i\pmod{n_i}\) with distinct \(n_i\) such that every integer satisfies exactly one of these congruence classes. This is true, and was proved independently by Mirsky and Newman and by Davenport and Rado; the Mirsky–Newman proof first appeared in [Er50]. See also [Er77c]. A `StrictCoveringSystem ℤ` is a finite family of congruence classes with distinct moduli \(n_i \geq 2\) covering \(\mathbb{Z}\); the trivial exact covering system consisting of the single class \(0 \pmod 1\) is therefore excluded.