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Ekhad","submitted_at":"2015-11-13T14:37:15Z","abstract_excerpt":"A set of arithmetical sequences $$ a_1\\, (\\bmod{ \\,\\, m_1}) \\quad, \\quad a_2 \\, (\\bmod{\\,\\, m_2}) \\quad, \\quad \\dots \\quad , \\quad a_k \\, (\\bmod{\\,\\,m_k}) \\quad \\quad , $$ with $$ m_1 \\leq m_2 \\leq \\dots \\leq m_k \\quad \\quad , $$ is called a {\\it disjoint covering system} (alias {\\it exact covering system}) if every positive integer belongs to {\\bf exactly} one of the sequences. Mirski, Newman, Davenport and Rado famously proved that the moduli can't all be distinct. 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