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Source. Theorem 1.1, PDF p. 2 of arXiv:math/0409279v2, the copy read for this card.

Statement

Let $\{a_s(n_s)\}_{s=1}^k$, where $k>1$ and each modulus $n_s$ is a positive integer, be a finite system of residue classes, and let

$$w(x)=\left|\{1\leq s\leq k:x\in a_s(n_s)\}\right|.$$

Suppose the range of $w$ is contained in one residue class with modulus $m$; the paper's residue classes $a(n)$ have moduli $n\in\mathbb Z^+$, so $m$ is a positive integer. For every $t\in\{1,\ldots,k\}$ such that

$$mn_t\nmid[n_1,\ldots,n_k],$$

there is an $s\in\{1,\ldots,k\}\setminus\{t\}$ for which $n_t\mid n_s$. Here $[n_1,\ldots,n_k]$ denotes the least common multiple.

Distinctness of all the moduli is not a hypothesis of this theorem.

Proof pointer. The source proves the theorem in Section 2, beginning on PDF p. 4, by evaluating the finite Fourier sum of the periodic covering function at suitable roots of unity. The proof was not reconstructed or independently checked here.

Bears on. It supplies Corollary 1.2, qualified context for Problem 7.