{"paper":{"title":"On a restricted linear congruence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Bruce M. Kapron, Khodakhast Bibak, Venkatesh Srinivasan","submitted_at":"2016-10-25T08:13:49Z","abstract_excerpt":"Let $b,n\\in \\mathbb{Z}$, $n\\geq 1$, and ${\\cal D}_1, \\ldots, {\\cal D}_{\\tau(n)}$ be all positive divisors of $n$. For $1\\leq l \\leq \\tau(n)$, define ${\\cal C}_l:=\\lbrace 1 \\leqslant x\\leqslant n \\; : \\; (x,n)={\\cal D}_l\\rbrace$. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence $x_1+\\cdots +x_k\\equiv b \\pmod{n}$, with $\\kappa_{l}=|\\lbrace x_1, \\ldots, x_k \\rbrace \\cap {\\cal C}_l|$, $1\\leq l \\leq \\tau(n)$, is \\begin{align*} \\frac{1}{n}\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07776","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}