{"id":160,"date":"2025-01-12T13:00:02","date_gmt":"2025-01-12T05:00:02","guid":{"rendered":"http:\/\/150.230.63.147\/blog\/?p=160"},"modified":"2025-01-12T13:00:02","modified_gmt":"2025-01-12T05:00:02","slug":"e-modular-stability-%e8%a7%a3%e6%9e%90%e6%80%9d%e7%b6%ad%e3%80%81%e6%95%b8%e8%ab%96%e3%80%81%e7%b5%84%e5%90%88","status":"publish","type":"post","link":"https:\/\/blog.seekrealthing.net\/blog\/?p=160","title":{"rendered":"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408)"},"content":{"rendered":"\n

Codeforce 1359 E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408)<\/h1>\n

\u4eca\u5929\u6211\u5011\u4f86\u770b\u770bCF1359E
\n\u984c\u76ee\u9023\u7d50<\/a><\/p>\n

\n

\u984c\u76ee<\/strong>
\n\u7565\uff0c\u8acb\u76f4\u63a5\u770b\u539f\u984c\u3002<\/p>\n<\/blockquote>\n

\u524d\u8a00<\/h3>\n

b<\/p>\n

\u60f3\u6cd5<\/h3>\n

\u9996\u5148\u6211\u5011\u89c0\u5bdf$k=2$\u7684\u60c5\u6cc1\uff0c\u56e0\u70ba\u7e3d\u611f\u89ba\u53ea\u8981\u5169\u500b\u6578\u5b57\u7684\u60c5\u6cc1\u660e\u77ad\u4e86\uff0c\u61c9\u8a72\u4e0d\u96e3\u63a8\u5ee3\u5230$k$\u500b\u6578\u5b57\u7684\u60c5\u6cc1\u3002
\nW.L.O.G. \u5047\u8a2d$1\\le a_1<a_2\\le n$\uff0c\u90a3\u9ebc\u6211\u5011\u6703\u6709\u4ee5\u4e0b\u7684\u5f0f\u5b50\uff1a$\\forall x\\in\\mathbb{N}_{\\ge0}, (x%a_1)%a_2=x%a_1\\textbf{=}(x%a_2)%a_1$\uff0c\u4e5f\u5c31\u662f\u8aaa$x=u\\cdot a_1+(x%a_1)=y\\cdot a_2+r\\cdot a_1+(x%a_1)$($u,y,r$\u662f\u4e00\u4e9b\u6578\u5b57)\u3002
\n\u6211\u5011\u731c\u60f3$a_1|a_2$\uff0c\u8981\u8b49\u660e\u9019\u4ef6\u4e8b\uff0c\u9996\u5148\u5047\u8a2d$a_1\\not|a_2$\u3002\u4e26\u4e14\u7531\u65bc$x$\u662f\u4efb\u610f\u975e\u8ca0\u6574\u6578\uff0c\u6211\u5011\u53ef\u4ee5\u53d6$x$\u4f7f\u5f97$x%a_1=0$\uff0c\u4e5f\u5c31\u662f\u8aaa$x=u\\cdot a_1=y\\cdot a_2+r\\cdot a_1$\uff0c\u4e26\u4e14$r\\cdot a_1<a_2$\u3002\u7136\u800c\u7531\u65bc$x$\u662f\u4efb\u610f\u7684\u975e\u8ca0\u6574\u6578\uff0c$u$\u4e5f\u5c31\u662f\u4efb\u610f\u7684\u975e\u8ca0\u6574\u6578\uff0c\u6240\u4ee5\u8aaa\u6211\u5011\u53ef\u4ee5\u53d6$u$\u4f7f\u5f97$u\\cdot a_1=y\\cdot a_2+r\\cdot a_1$\u4e26\u4e14$(u+1)\\cdot a_1=(y+1)\\cdot a_2+r\u2019\\cdot a_1$\uff0c\u5169\u5f0f\u76f8\u6e1b\u5f97\u5230$(1-(r-r\u2019))a_1=a_2$\uff0c\u7136\u800c$r,r\u2019\\in\\mathbb{N}$\uff0c\u56e0\u6b64\u548c\u5047\u8a2d\u76f8\u77db\u76fe\uff0c\u5b9a\u7406\u5f97\u8b49\uff0c$a_1|a_2$\u3002
\n\u90a3\u9ebc\u63a5\u4e0b\u4f86\u8b49\u660e\uff0c\u5047\u8a2d$a_1$\u662f\u88e1\u9762\u6700\u5c0f\u7684\u6578\u5b57\uff0c$a_1|a_i\\forall i\\iff\u7b26\u5408\u689d\u4ef6$\u3002
\n$(\\Rightarrow):$\u7531\u65bc\u5c0d\u65bc\u4efb\u4f55\u7684\u53d6\u6a21\u6392\u5217\uff0c\u6211\u5011\u7e3d\u662f\u53ef\u4ee5\u5148\u5c0d$a_1$\u53d6\u6a21(\u7528\u5230\u4e0a\u9762\u8b49\u660e\u7684$k=2$\u7684\u60c5\u6cc1)\uff0c\u56e0\u6b64\u5f97\u8b49\u3002
\n$(\\Leftarrow):$\u5047\u8a2d$a_1\\not|a_j$\u5c0d\u65bc\u67d0\u500b$j$\uff0c\u90a3\u9ebc$x%a_1%a_j%\u2026\\neq x%a_j%a_1%\u2026$\u5c0d\u65bc\u67d0\u500b$x$\u3002<\/p>\n

\u56e0\u6b64\u6211\u5011\u53ea\u8981\u5148\u6c7a\u5b9a\u4e86$a_1$\uff0c\u63a5\u4e0b\u4f86\u7684\u6578\u5b57\u90fd\u6703\u662f$a_1$\u7684\u500d\u6578\uff0c\u7528\u7d44\u5408\u6578\u8a08\u7b97\u53ef\u80fd\u6027\u5373\u53ef\u3002<\/p>\n

\u7a0b\u5f0f\u78bc:<\/h3>\n
const int _n=5e5+10;  \nint t,n,k,fac[_n],inv[_n];;  \nll ans;  \nint C(int m,int n){  \n  if(m<n)return 0;  \n  if(m<mod and n<mod)return 1ll*fac[m]*inv[n]%mod*inv[m-n]%mod;  \n  return 1ll*C(m\/mod,n\/mod)*C(m%mod,n%mod)%mod;  \n}  \nvoid genInv(){  \n  fac[0]=1;rep(i,1,n+1)fac[i]=1ll*fac[i-1]*i%mod;=  \n  inv[n]=powmod(fac[n],mod-2);per(i,0,n)inv[i]=1ll*inv[i+1]*(i+1)%mod;  \n}  \nmain(void) {ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);  \n  cin>>n>>k;genInv();rep(i,1,n\/k+1)ans=(ans+1ll*C(n\/i-1,k-1))%mod;  \n  cout<<ans<<'\\n';  \n  return 0;  \n}  \n<\/code><\/pre>\n
\u6a19\u982d\u3001\u6a21\u677f\u8acb\u9edeSubmission\u770b
\nSubmission<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[3],"tags":[10,9,8,21,6],"class_list":["post-160","post","type-post","status-publish","format-standard","hentry","category-ojcode","tag-codeforce","tag-ojcode","tag-8","tag-21","tag-6"],"jetpack_publicize_connections":[],"yoast_head":"\nE. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408) - Logic Formatting<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\" \/>\n<meta property=\"og:locale\" content=\"zh_TW\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408) - Logic Formatting\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\" \/>\n<meta property=\"og:site_name\" content=\"Logic Formatting\" \/>\n<meta property=\"article:published_time\" content=\"2025-01-12T05:00:02+00:00\" \/>\n<meta name=\"author\" content=\"jeff\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005:\" \/>\n\t<meta name=\"twitter:data1\" content=\"jeff\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9810\u4f30\u95b1\u8b80\u6642\u9593\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 \u5206\u9418\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160#article\",\"isPartOf\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\"},\"author\":{\"name\":\"jeff\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/233e80a0fee3d4706f067f2e88c0b454\"},\"headline\":\"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408)\",\"datePublished\":\"2025-01-12T05:00:02+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\"},\"wordCount\":140,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#organization\"},\"keywords\":[\"codeforce\",\"ojcode\",\"\u601d\u7dad\",\"\u6578\u8ad6\",\"\u7d44\u5408\"],\"articleSection\":[\"ojcode\"],\"inLanguage\":\"zh-TW\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/blog.seekrealthing.net\/blog\/?p=160#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\",\"url\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\",\"name\":\"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408) - Logic Formatting\",\"isPartOf\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#website\"},\"datePublished\":\"2025-01-12T05:00:02+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160#breadcrumb\"},\"inLanguage\":\"zh-TW\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/blog.seekrealthing.net\/blog\/?p=160\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/?p=160#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"\u9996\u9801\",\"item\":\"https:\/\/blog.seekrealthing.net\/blog\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#website\",\"url\":\"https:\/\/blog.seekrealthing.net\/blog\/\",\"name\":\"Logic Formatting\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/blog.seekrealthing.net\/blog\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"zh-TW\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#organization\",\"name\":\"Logic Formatting\",\"url\":\"https:\/\/blog.seekrealthing.net\/blog\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-TW\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/android-chrome-512x512-1.png\",\"contentUrl\":\"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/android-chrome-512x512-1.png\",\"width\":512,\"height\":512,\"caption\":\"Logic Formatting\"},\"image\":{\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/233e80a0fee3d4706f067f2e88c0b454\",\"name\":\"jeff\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-TW\",\"@id\":\"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f9b571b28c252728386f0002b35d9cb9a9028f740b0f9ca5ad678e7eb21c4857?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f9b571b28c252728386f0002b35d9cb9a9028f740b0f9ca5ad678e7eb21c4857?s=96&d=mm&r=g\",\"caption\":\"jeff\"},\"sameAs\":[\"http:\/\/150.230.63.147\/blog\"],\"url\":\"https:\/\/blog.seekrealthing.net\/blog\/?author=1\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408) - Logic Formatting","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/blog.seekrealthing.net\/blog\/?p=160","og_locale":"zh_TW","og_type":"article","og_title":"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408) - Logic Formatting","og_url":"https:\/\/blog.seekrealthing.net\/blog\/?p=160","og_site_name":"Logic Formatting","article_published_time":"2025-01-12T05:00:02+00:00","author":"jeff","twitter_card":"summary_large_image","twitter_misc":{"\u4f5c\u8005:":"jeff","\u9810\u4f30\u95b1\u8b80\u6642\u9593":"1 \u5206\u9418"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/blog.seekrealthing.net\/blog\/?p=160#article","isPartOf":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/?p=160"},"author":{"name":"jeff","@id":"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/233e80a0fee3d4706f067f2e88c0b454"},"headline":"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408)","datePublished":"2025-01-12T05:00:02+00:00","mainEntityOfPage":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/?p=160"},"wordCount":140,"commentCount":0,"publisher":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/#organization"},"keywords":["codeforce","ojcode","\u601d\u7dad","\u6578\u8ad6","\u7d44\u5408"],"articleSection":["ojcode"],"inLanguage":"zh-TW","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/blog.seekrealthing.net\/blog\/?p=160#respond"]}]},{"@type":"WebPage","@id":"https:\/\/blog.seekrealthing.net\/blog\/?p=160","url":"https:\/\/blog.seekrealthing.net\/blog\/?p=160","name":"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408) - Logic Formatting","isPartOf":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/#website"},"datePublished":"2025-01-12T05:00:02+00:00","breadcrumb":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/?p=160#breadcrumb"},"inLanguage":"zh-TW","potentialAction":[{"@type":"ReadAction","target":["https:\/\/blog.seekrealthing.net\/blog\/?p=160"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/blog.seekrealthing.net\/blog\/?p=160#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"\u9996\u9801","item":"https:\/\/blog.seekrealthing.net\/blog"},{"@type":"ListItem","position":2,"name":"E. Modular Stability \u89e3\u6790(\u601d\u7dad\u3001\u6578\u8ad6\u3001\u7d44\u5408)"}]},{"@type":"WebSite","@id":"https:\/\/blog.seekrealthing.net\/blog\/#website","url":"https:\/\/blog.seekrealthing.net\/blog\/","name":"Logic Formatting","description":"","publisher":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/blog.seekrealthing.net\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"zh-TW"},{"@type":"Organization","@id":"https:\/\/blog.seekrealthing.net\/blog\/#organization","name":"Logic Formatting","url":"https:\/\/blog.seekrealthing.net\/blog\/","logo":{"@type":"ImageObject","inLanguage":"zh-TW","@id":"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/logo\/image\/","url":"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/android-chrome-512x512-1.png","contentUrl":"https:\/\/blog.seekrealthing.net\/blog\/wp-content\/uploads\/2025\/02\/android-chrome-512x512-1.png","width":512,"height":512,"caption":"Logic Formatting"},"image":{"@id":"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/233e80a0fee3d4706f067f2e88c0b454","name":"jeff","image":{"@type":"ImageObject","inLanguage":"zh-TW","@id":"https:\/\/blog.seekrealthing.net\/blog\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f9b571b28c252728386f0002b35d9cb9a9028f740b0f9ca5ad678e7eb21c4857?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f9b571b28c252728386f0002b35d9cb9a9028f740b0f9ca5ad678e7eb21c4857?s=96&d=mm&r=g","caption":"jeff"},"sameAs":["http:\/\/150.230.63.147\/blog"],"url":"https:\/\/blog.seekrealthing.net\/blog\/?author=1"}]}},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=\/wp\/v2\/posts\/160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=160"}],"version-history":[{"count":1,"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=\/wp\/v2\/posts\/160\/revisions"}],"predecessor-version":[{"id":161,"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=\/wp\/v2\/posts\/160\/revisions\/161"}],"wp:attachment":[{"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.seekrealthing.net\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}