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Claim. For moduli $n_1<\cdots<n_r$ and any residues $a_i$, the density of the integers covered by the classes $a_i\pmod{n_i}$ is at least

$$\sum_i\frac{1}{n_i}-\sum_{i<j}\frac{1}{[n_i,n_j]} +\sum_{i<j<k}\frac{1}{[n_i,n_j,n_k]}-\cdots,$$

where $[\cdots]$ is the least common multiple, and this value is attained when all the $a_i$ are equal. The answer to the second question of Problem 278 is therefore yes: the minimum covered density, equivalently the maximum uncovered density, is achieved by equal residues. When the residues agree, every subfamily of classes meets in one class modulo its least common multiple, so inclusion-exclusion evaluates the covered density exactly as the displayed sum; that the sum is a lower bound for every residue choice is the paper's theorem, Simpson's Lemma 2.3, as the site's commentary records it and as Cambie's note cites it. The result is in R. J. Simpson, Exact coverings of the integers by arithmetic progressions, Discrete Math. 59 (1986), 181--190, cited as [Si86] on the problem page and as Lemma 2.3 of that paper by Cambie's note (library card). The paper is not held in the library; the statement is recorded as the site's commentary and Cambie's note give it, and its proof is not checked in this corpus. The same theorem appears, credited to C. A. Rogers, in Halberstam and Roth's Sequences, Vol. I (Clarendon Press, Oxford, 1966), pp. 242--244, as Filaseta, Ford, Konyagin, Pomerance and Yu cite it (library card); a comment of 2026-03-03 on the problem's discussion thread points to the 1983 Springer reissue of the book. That earlier publication is recorded on its own claim page; this page keeps the published author the site credits as its claimant.

Covers. The second question alone: equal residues minimize the covered density. The first question, the value of the maximum covered density for a given set of moduli, is not addressed; its pending claims are Cambie's, Onishi's and Schroeder's.

Depends on. Nothing in this wiki: the bound is the paper's own.

Acceptance. Refereed: Discrete Mathematics 59 (1986), no. 1--2, 181--190, the DOI linked above; the Crossref record dates the issue to April 1986 without a day, so the page is named by the first of that month. The claim is accepted on the refereed paper alone. The site's curator, Thomas F. Bloom, states in the problem page's commentary that Simpson's observation settles the second question, and Cambie's note restates the result as settled; but the site labels the problem OPEN (page last edited 20 January 2026), and commentary on a problem the site labels OPEN is not acceptance, so the curator's remark is context and not reviewed evidence. Nothing on this page rests on a review by this project.