{"id":24716,"date":"2025-05-19T23:09:52","date_gmt":"2025-05-19T20:09:52","guid":{"rendered":"https:\/\/w3.bilkent.edu.tr\/en\/?page_id=24716"},"modified":"2025-07-08T13:09:31","modified_gmt":"2025-07-08T10:09:31","slug":"problem-of-the-month-may-2025","status":"publish","type":"page","link":"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-may-2025\/","title":{"rendered":"Problem of the Month \u2013 May 2025"},"content":{"rendered":"<p>Let <em>n<\/em> be a positive integer and S<sub><em>n<\/em><\/sub> be the set of all positive integers not exceeding <em>n<\/em> and relatively prime to <em>n<\/em>. Let f(<em>n<\/em>) be the smallest positive integer for which the set S<sub><em>n<\/em><\/sub> can be partitioned into f(<em>n<\/em>) disjoint subsets, such that each of these subsets is an arithmetic progression. Show that there are infinitely many pairs (<em>a, b<\/em>) such that <em>a, b<\/em> &gt; 2025, <em>a<\/em>|<em>b<\/em> and f(<em>a<\/em>) 6 |f(<em>b<\/em>).<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Correct Solutions by,<\/strong><\/p>\n<ul>\n<li>Toshihiro Shimizu Kawasaki, Japan<\/li>\n<li>Mahmut Semih Ba\u015f\u00a0\u0130stanbul Technical University<\/li>\n<li>Ramazan Kara\u015fahin\u00a0Ankara<\/li>\n<li>Hasan Z\u00fcbeyr Demir\u00a0Ankara<\/li>\n<li>Magnus Jakobsson Lund, Sweden<\/li>\n<li>\u00d6zg\u00fcr Soysal\u00a0Bilkent University<\/li>\n<li>Murat Be\u015fli\u00a0Politecnico di Torino, Italy<\/li>\n<li>Arda Karahan\u00a0Trabzon Fen Lisesi<\/li>\n<li>Ananda Raidu\u00a0Rajahmundry, India<\/li>\n<li>Serhat G\u00fcldo\u011fan\u00a0Ko\u00e7 University<\/li>\n<\/ul>\n<p><strong>Solution <a href=\"https:\/\/math.bilkent.edu.tr\/Soru\/2505c.pdf\">2505c.pdf (bilkent.edu.tr)<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let n be a positive integer and Sn be the set of all positive integers not exceeding n and relatively prime to n. Let f(n) be the smallest positive integer for which the set Sn can be partitioned into f(n) disjoint subsets, such that each of these subsets is an [\u2026] <\/p>\n<div class=\"clear\"><\/div>\n<p><a class=\"more_link clearfix\" href=\"https:\/\/w3.bilkent.edu.tr\/en\/problem-of-the-month-may-2025\/\" rel=\"nofollow\">Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-24716","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/24716","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/comments?post=24716"}],"version-history":[{"count":4,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/24716\/revisions"}],"predecessor-version":[{"id":25052,"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/pages\/24716\/revisions\/25052"}],"wp:attachment":[{"href":"https:\/\/w3.bilkent.edu.tr\/en\/wp-json\/wp\/v2\/media?parent=24716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}