CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>D. R. Kaprekar - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"c4c6696a-0970-4511-8515-4266c1626713","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"D._R._Kaprekar","wgTitle":"D. R. Kaprekar","wgCurRevisionId":1256679320,"wgRevisionId":1256679320,"wgArticleId":1664575,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 Hindi-language sources (hi)","Articles with short description","Short description is different from Wikidata","EngvarB from September 2014","Use dmy dates from September 2014","Articles with hCards","Articles containing Marathi-language text","All articles with unsourced statements","Articles with unsourced statements from September 2024","Articles with unsourced statements from May 2020","1905 births","1986 deaths","20th-century Indian mathematicians","Indian number theorists" ,"Recreational mathematicians","Magic squares","People from Thane district","Scientists from Maharashtra"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"D._R._Kaprekar","wgRelevantArticleId":1664575,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage", "wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q560998","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=[ "ext.cite.ux-enhancements","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/en/b/b1/D._R._Kaprekar.gif"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1566"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/en/b/b1/D._R._Kaprekar.gif"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="1044"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="835"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="D. R. Kaprekar - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/D._R._Kaprekar"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=D._R._Kaprekar&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/D._R._Kaprekar"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-D_R_Kaprekar rootpage-D_R_Kaprekar skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=D.+R.+Kaprekar" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=D.+R.+Kaprekar" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=D.+R.+Kaprekar" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=D.+R.+Kaprekar" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Education_and_work" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Education_and_work"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Education and work</span> </div> </a> <ul id="toc-Education_and_work-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discoveries" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Discoveries"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Discoveries</span> </div> </a> <button aria-controls="toc-Discoveries-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Discoveries subsection</span> </button> <ul id="toc-Discoveries-sublist" class="vector-toc-list"> <li id="toc-Kaprekar's_Constant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kaprekar's_Constant"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Kaprekar's Constant</span> </div> </a> <ul id="toc-Kaprekar's_Constant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kaprekar_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kaprekar_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kaprekar number</span> </div> </a> <ul id="toc-Kaprekar_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Devlali_or_self_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Devlali_or_self_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Devlali or self number</span> </div> </a> <ul id="toc-Devlali_or_self_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harshad_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harshad_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Harshad number</span> </div> </a> <ul id="toc-Harshad_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Demlo_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Demlo_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Demlo number</span> </div> </a> <ul id="toc-Demlo_number-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">D. R. Kaprekar</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 25 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-25" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">25 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BF._%E0%A6%86%E0%A6%B0._%E0%A6%95%E0%A6%BE%E0%A6%AA%E0%A6%B0%E0%A7%87%E0%A6%95%E0%A6%BE%E0%A6%B0" title="ডি. আর. কাপরেকার – Bangla" lang="bn" hreflang="bn" data-title="ডি. আর. কাপরেকার" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/D._R._Kaprekar" title="D. R. Kaprekar – Catalan" lang="ca" hreflang="ca" data-title="D. R. Kaprekar" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dattathreya_Ramachandra_Kaprekar" title="Dattathreya Ramachandra Kaprekar – German" lang="de" hreflang="de" data-title="Dattathreya Ramachandra Kaprekar" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dattatreya_Ramachandra_Kaprekar" title="Dattatreya Ramachandra Kaprekar – Spanish" lang="es" hreflang="es" data-title="Dattatreya Ramachandra Kaprekar" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%DB%8C._%D8%A2%D8%B1._%DA%A9%D8%A7%D9%BE%D8%B1%DA%A9%D8%A7%D8%B1" title="دی. آر. کاپرکار – Persian" lang="fa" hreflang="fa" data-title="دی. آر. کاپرکار" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Dattatreya_Ramachandra_Kaprekar" title="Dattatreya Ramachandra Kaprekar – French" lang="fr" hreflang="fr" data-title="Dattatreya Ramachandra Kaprekar" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Dattatreya_Ramachandra_Kaprekar" title="Dattatreya Ramachandra Kaprekar – Galician" lang="gl" hreflang="gl" data-title="Dattatreya Ramachandra Kaprekar" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%AF_%E0%A4%B0%E0%A4%BE%E0%A4%AE%E0%A4%9A%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A5%8D%E0%A4%B0_%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%AA%E0%A5%87%E0%A4%95%E0%A4%B0" title="दत्तात्रय रामचन्द्र कर्पेकर – Hindi" lang="hi" hreflang="hi" data-title="दत्तात्रय रामचन्द्र कर्पेकर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dattatreya_Ramachandra_Kaprekar" title="Dattatreya Ramachandra Kaprekar – Italian" lang="it" hreflang="it" data-title="Dattatreya Ramachandra Kaprekar" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%90%D7%98%D7%90%D7%A8%D7%99%D7%99%D7%94_%D7%A8%D7%9E%D7%90%D7%A6%27%D7%90%D7%A0%D7%93%D7%A8%D7%94_%D7%A7%D7%A4%D7%A8%D7%A7%D7%A8" title="דאטארייה רמאצ'אנדרה קפרקר – Hebrew" lang="he" hreflang="he" data-title="דאטארייה רמאצ'אנדרה קפרקר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A6%E0%B2%A4%E0%B3%8D%E0%B2%A4%E0%B2%BE%E0%B2%A4%E0%B3%8D%E0%B2%B0%E0%B3%87%E0%B2%AF_%E0%B2%B0%E0%B2%BE%E0%B2%AE%E0%B2%9A%E0%B2%82%E0%B2%A6%E0%B3%8D%E0%B2%B0_%E0%B2%95%E0%B2%BE%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B3%86%E0%B2%95%E0%B2%B0%E0%B3%8D" title="ದತ್ತಾತ್ರೇಯ ರಾಮಚಂದ್ರ ಕಾಪ್ರೆಕರ್ – Kannada" lang="kn" hreflang="kn" data-title="ದತ್ತಾತ್ರೇಯ ರಾಮಚಂದ್ರ ಕಾಪ್ರೆಕರ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Kaprekar" title="Kaprekar – Haitian Creole" lang="ht" hreflang="ht" data-title="Kaprekar" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A1%E0%B4%BF.%E0%B4%86%E0%B5%BC._%E0%B4%95%E0%B4%BE%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B5%87%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B5%BC" title="ഡി.ആർ. കാപ്രേക്കർ – Malayalam" lang="ml" hreflang="ml" data-title="ഡി.ആർ. കാപ്രേക്കർ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%A6%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%AF_%E0%A4%B0%E0%A4%BE%E0%A4%AE%E0%A4%9A%E0%A4%82%E0%A4%A6%E0%A5%8D%E0%A4%B0_%E0%A4%95%E0%A4%BE%E0%A4%AA%E0%A4%B0%E0%A5%87%E0%A4%95%E0%A4%B0" title="दत्तात्रेय रामचंद्र कापरेकर – Marathi" lang="mr" hreflang="mr" data-title="दत्तात्रेय रामचंद्र कापरेकर" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Dattathreya_Ramachandra_Kaprekar" title="Dattathreya Ramachandra Kaprekar – Dutch" lang="nl" hreflang="nl" data-title="Dattathreya Ramachandra Kaprekar" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%80%E3%83%83%E3%82%BF%E3%83%88%E3%83%AA%E3%83%A4%E3%83%BB%E3%83%A9%E3%83%A0%E3%83%81%E3%83%A3%E3%83%B3%E3%83%89%E3%83%A9%E3%83%BB%E3%82%AB%E3%83%97%E3%83%AC%E3%82%AB%E3%83%AB" title="ダッタトリヤ・ラムチャンドラ・カプレカル – Japanese" lang="ja" hreflang="ja" data-title="ダッタトリヤ・ラムチャンドラ・カプレカル" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dattathreya_Ramachandra_Kaprekar" title="Dattathreya Ramachandra Kaprekar – Polish" lang="pl" hreflang="pl" data-title="Dattathreya Ramachandra Kaprekar" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/D._R._Kaprekar" title="D. R. Kaprekar – Portuguese" lang="pt" hreflang="pt" data-title="D. R. Kaprekar" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BF%D1%80%D0%B5%D0%BA%D0%B0%D1%80,_%D0%94%D0%B0%D1%82%D1%82%D0%B0%D1%80%D0%B0%D1%8F_%D0%A0%D0%B0%D0%BC%D1%87%D0%B0%D0%BD%D0%B4%D1%80%D0%B0" title="Капрекар, Даттарая Рамчандра – Russian" lang="ru" hreflang="ru" data-title="Капрекар, Даттарая Рамчандра" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%A6%E0%A4%A4%E0%A5%8D%E2%80%8D%E0%A4%A4%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%AF_%E0%A4%B0%E0%A4%BE%E0%A4%AE%E0%A4%9A%E0%A4%82%E0%A4%A6%E0%A5%8D%E0%A4%B0_%E0%A4%95%E0%A4%BE%E0%A4%AA%E0%A4%B0%E0%A5%87%E0%A4%95%E0%A4%B0" title="दत्‍तात्रेय रामचंद्र कापरेकर – Sanskrit" lang="sa" hreflang="sa" data-title="दत्‍तात्रेय रामचंद्र कापरेकर" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/D._R._Kaprekar" title="D. R. Kaprekar – Finnish" lang="fi" hreflang="fi" data-title="D. R. Kaprekar" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/D.R._Kaprekar" title="D.R. Kaprekar – Swedish" lang="sv" hreflang="sv" data-title="D.R. Kaprekar" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%B0%E0%AE%BE%E0%AE%AF_%E0%AE%B0%E0%AE%BE%E0%AE%AE%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B0_%E0%AE%95%E0%AE%AA%E0%AF%8D%E0%AE%B0%E0%AF%87%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B0%E0%AF%8D" title="தத்தராய ராமச்சந்திர கப்ரேக்கர் – Tamil" lang="ta" hreflang="ta" data-title="தத்தராய ராமச்சந்திர கப்ரேக்கர்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A6%E0%B0%A4%E0%B1%8D%E0%B0%A4%E0%B0%BE%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B1%87%E0%B0%AF_%E0%B0%B0%E0%B0%BE%E0%B0%AE%E0%B0%9A%E0%B0%82%E0%B0%A6%E0%B1%8D%E0%B0%B0_%E0%B0%95%E0%B0%AA%E0%B1%8D%E0%B0%B0%E0%B1%87%E0%B0%95%E0%B0%B0%E0%B1%8D" title="దత్తాత్రేయ రామచంద్ర కప్రేకర్ – Telugu" lang="te" hreflang="te" data-title="దత్తాత్రేయ రామచంద్ర కప్రేకర్" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kaprekar" title="Kaprekar – Turkish" lang="tr" hreflang="tr" data-title="Kaprekar" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q560998#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/D._R._Kaprekar" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:D._R._Kaprekar" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/D._R._Kaprekar"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/D._R._Kaprekar"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/D._R._Kaprekar" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/D._R._Kaprekar" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&oldid=1256679320" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=D._R._Kaprekar&id=1256679320&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FD._R._Kaprekar"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FD._R._Kaprekar"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=D._R._Kaprekar&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=D._R._Kaprekar&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q560998" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Indian recreational mathematician (1905–1986)</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">D. R. Kaprekar</div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:D._R._Kaprekar.gif" class="mw-file-description"><img alt="D. R. Kaprekar" src="//upload.wikimedia.org/wikipedia/en/b/b1/D._R._Kaprekar.gif" decoding="async" width="174" height="227" class="mw-file-element" data-file-width="174" data-file-height="227" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><div style="display:inline" class="nickname">D.R. Kaprekar</div><br /><span style="display:none">(<span class="bday">1905-01-17</span>)</span>17 January 1905<br /><div style="display:inline" class="birthplace"><a href="/wiki/Dahanu" title="Dahanu">Dahanu</a>, <a href="/wiki/Bombay_Presidency" title="Bombay Presidency">Bombay Presidency</a>, <a href="/wiki/British_Raj" title="British Raj">India</a></div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">1986<span style="display:none">(1986-00-00)</span> (aged 80–81)<br /><div style="display:inline" class="deathplace"><a href="/wiki/Devlali" class="mw-redirect" title="Devlali">Devlali</a>, Maharashtra, India</div></td></tr><tr><th scope="row" class="infobox-label">Occupation</th><td class="infobox-data role">School teacher</td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data">Contributions to <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a></td></tr></tbody></table> <p><b>Dattatreya Ramchandra Kaprekar</b> (<a href="/wiki/Marathi_language" title="Marathi language">Marathi</a>: <span lang="mr">दत्तात्रेय रामचंद्र कापरेकर</span>; 17 January 1905 – 1986) was an Indian <a href="/wiki/Recreational_mathematician" class="mw-redirect" title="Recreational mathematician">recreational mathematician</a> who described several <a href="/wiki/Template:Classes_of_natural_numbers" title="Template:Classes of natural numbers">classes of natural numbers</a> including the <a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar</a>, <a href="/wiki/Harshad_number" title="Harshad number">harshad</a> and <a href="/wiki/Self_number" title="Self number">self</a> numbers and discovered the <a href="/wiki/Kaprekar%27s_constant" class="mw-redirect" title="Kaprekar's constant">Kaprekar's constant</a>, named after him.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles.<sup id="cite_ref-mactutor_2-0" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Education_and_work">Education and work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=1" title="Edit section: Education and work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kaprekar received his secondary school education in <a href="/wiki/Thane" title="Thane">Thane</a> and studied at <a href="/wiki/Pune_University" class="mw-redirect" title="Pune University">Fergusson College</a> in <a href="/wiki/Pune" title="Pune">Pune</a>. In 1927, he won the Wrangler R. P. Paranjpye Mathematical Prize for an original piece of work in mathematics.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>He attended the <a href="/wiki/University_of_Mumbai" title="University of Mumbai">University of Mumbai</a>, receiving his bachelor's degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at the government junior school in Devlali <a href="/wiki/Maharashtra" title="Maharashtra">Maharashtra</a>, India. Cycling from place to place he also tutored private students with unconventional methods, cheerfully sitting by a river and "thinking of theorems". He published extensively, writing about such topics as <a href="/wiki/Recurring_decimal" class="mw-redirect" title="Recurring decimal">recurring decimals</a>, <a href="/wiki/Magic_squares" class="mw-redirect" title="Magic squares">magic squares</a>, and integers with special properties.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Discoveries">Discoveries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=2" title="Edit section: Discoveries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In addition to the <a href="/wiki/Kaprekar%27s_constant" class="mw-redirect" title="Kaprekar's constant">Kaprekar's constant</a> and the <a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar numbers</a> which were named after him, he also described <a href="/wiki/Self_number" title="Self number">self numbers</a> or <i>Devlali numbers</i>, the <a href="/wiki/Harshad_number" title="Harshad number">harshad numbers</a> and <a href="/wiki/Demlo_number" class="mw-redirect" title="Demlo number">Demlo numbers</a>. He also constructed certain types of magic squares related to the Copernicus magic square.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Initially his ideas were not taken seriously by Indian mathematicians, and his results were published largely in low-level mathematics journals or privately published, but international fame arrived when <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> wrote about Kaprekar in his March 1975 column of <i>Mathematical Games</i> for <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. A description of <a href="/wiki/Kaprekar%27s_constant" class="mw-redirect" title="Kaprekar's constant">Kaprekar's constant</a>, without mention of Kaprekar, appears in the children's book <i>The I Hate Mathematics Book</i>, by <a href="/wiki/Marilyn_Burns_(mathematics_educator)" title="Marilyn Burns (mathematics educator)">Marilyn Burns</a>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> published in 1975. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered.<sup id="cite_ref-mactutor_2-1" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kaprekar's_Constant"><span id="Kaprekar.27s_Constant"></span>Kaprekar's Constant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=3" title="Edit section: Kaprekar's Constant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/D_R_Kaprekar" class="mw-redirect" title="D R Kaprekar">D R Kaprekar</a></div> <p>In 1955, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> He showed that 6174 is reached in the end as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have: </p> <dl><dd>4321 − 1234 = 3087, then</dd> <dd>8730 − 0378 = 8352, and</dd> <dd>8532 − 2358 = 6174.</dd></dl> <p>Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations. </p><p>A similar constant for 3 digits is <a href="/wiki/495_(number)" title="495 (number)">495</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the <a href="/wiki/Kaprekar%27s_routine" title="Kaprekar's routine">Kaprekar's routine</a> algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kaprekar_number">Kaprekar number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=4" title="Edit section: Kaprekar number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kaprekar_number" title="Kaprekar number">Kaprekar number</a></div> <p>Another class of numbers Kaprekar described are Kaprekar numbers.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 45<sup>2</sup>=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 100<sup>2</sup>=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation. </p><p>Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, ..., are (sequence <span class="nowrap external"><a href="//oeis.org/A006886" class="extiw" title="oeis:A006886">A006886</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <table class="wikitable"> <tbody><tr> <th>Number</th> <th>Square</th> <th>Decomposition </th></tr> <tr> <td>703</td> <td>703² = 494209</td> <td>494+209 = 703 </td></tr> <tr> <td>2728</td> <td>2728² = 7441984</td> <td>744+1984 = 2728 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Devlali_or_self_number">Devlali or self number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=5" title="Edit section: Devlali or self number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Self_number" title="Self number">Self number</a></div> <p>In 1963, Kaprekar defined the property which has come to be known as self numbers,<sup id="cite_ref-devlali_11-0" class="reference"><a href="#cite_note-devlali-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> as the integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to have been his preferred designation,<sup id="cite_ref-devlali_11-1" class="reference"><a href="#cite_note-devlali-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> the term "self number" is more widespread. Sometimes these are also designated <i>Colombian number</i>s after a later designation. </p> <div class="mw-heading mw-heading3"><h3 id="Harshad_number">Harshad number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=6" title="Edit section: Harshad number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Harshad_number" title="Harshad number">Harshad number</a></div> <p>Kaprekar also described the <a href="/wiki/Harshad_number" title="Harshad number">harshad numbers</a> which he named harshad, meaning "giving joy" (<a href="/wiki/Sanskrit" title="Sanskrit">Sanskrit</a> <i>harsha</i>, joy <i>+da</i> taddhita pratyaya, <a href="/wiki/Causative" title="Causative">causative</a>); these are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a harshad number. These were later also called <i>Niven numbers</i> after 1977 lecture on these by the Canadian mathematician <a href="/wiki/Ivan_M._Niven" title="Ivan M. Niven">Ivan M. Niven</a>. Numbers which are harshad in all bases (only 1, 2, 4, and 6) are called <i>all-harshad numbers</i>. Much work has been done on harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2020)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Demlo_number">Demlo number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=7" title="Edit section: Demlo number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kaprekar also studied the <a href="/wiki/Demlo_number" class="mw-redirect" title="Demlo number">Demlo numbers</a>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> name of which was derived from the name of a train station Demlo (now called <a href="/wiki/Dombivli_railway_station" title="Dombivli railway station">Dombivili</a>) 30 miles from Bombay on the then <a href="/wiki/G._I._P._Railway" class="mw-redirect" title="G. I. P. Railway">G. I. P. Railway</a> where he had the idea of studying them.<sup id="cite_ref-mactutor_2-2" class="reference"><a href="#cite_note-mactutor-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The best known of these are the Wonderful Demlo numbers 1, 121, 12321, 1234321, ..., which are the squares of the <a href="/wiki/Repunit" title="Repunit">repunits</a> 1, 11, 111,1111, ....<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Prahalad_Chunnilal_Vaidya" title="Prahalad Chunnilal Vaidya">Prahalad Chunnilal Vaidya</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1 cs1-prop-foreign-lang-source"><a rel="nofollow" class="external text" href="https://www.aajtak.in/education/knowledge/story/indian-mathematician-dattatreya-ramchandra-kaprekar-who-invented-magical-6174-number-1616954-2023-01-17">"क्‍या आप जानते हैं जादुई नंबर 6174 की पहेली? इस भारतीय गणितज्ञ ने की खोज"</a>. <i>आज तक</i> (in Hindi). 17 January 2023<span class="reference-accessdate">. Retrieved <span class="nowrap">13 October</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=%E0%A4%86%E0%A4%9C+%E0%A4%A4%E0%A4%95&rft.atitle=%E0%A4%95%E0%A5%8D%E2%80%8D%E0%A4%AF%E0%A4%BE+%E0%A4%86%E0%A4%AA+%E0%A4%9C%E0%A4%BE%E0%A4%A8%E0%A4%A4%E0%A5%87+%E0%A4%B9%E0%A5%88%E0%A4%82+%E0%A4%9C%E0%A4%BE%E0%A4%A6%E0%A5%81%E0%A4%88+%E0%A4%A8%E0%A4%82%E0%A4%AC%E0%A4%B0+6174+%E0%A4%95%E0%A5%80+%E0%A4%AA%E0%A4%B9%E0%A5%87%E0%A4%B2%E0%A5%80%3F+%E0%A4%87%E0%A4%B8+%E0%A4%AD%E0%A4%BE%E0%A4%B0%E0%A4%A4%E0%A5%80%E0%A4%AF+%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%9C%E0%A5%8D%E0%A4%9E+%E0%A4%A8%E0%A5%87+%E0%A4%95%E0%A5%80+%E0%A4%96%E0%A5%8B%E0%A4%9C&rft.date=2023-01-17&rft_id=https%3A%2F%2Fwww.aajtak.in%2Feducation%2Fknowledge%2Fstory%2Findian-mathematician-dattatreya-ramchandra-kaprekar-who-invented-magical-6174-number-1616954-2023-01-17&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-mactutor-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-mactutor_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mactutor_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-mactutor_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Kaprekar.html">"D. R. Kaprekar"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=D.+R.+Kaprekar&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FKaprekar.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDilip_M._Salwi2005" class="citation web cs1">Dilip M. Salwi (24 January 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071116164833/http://www.4to40.com/legends/index.asp?id=142">"Dattaraya Ramchandra Kaprekar"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.4to40.com/legends/index.asp?id=142">the original</a> on 16 November 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">30 November</span> 2007</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dattaraya+Ramchandra+Kaprekar&rft.date=2005-01-24&rft.au=Dilip+M.+Salwi&rft_id=http%3A%2F%2Fwww.4to40.com%2Flegends%2Findex.asp%3Fid%3D142&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAthmaraman2004" class="citation book cs1">Athmaraman, R. (2004). <i>The Wonder World of Kaprekar Numbers</i>. Chennai (India): The Association of Mathematics Teachers of India.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Wonder+World+of+Kaprekar+Numbers&rft.place=Chennai+%28India%29&rft.pub=The+Association+of+Mathematics+Teachers+of+India&rft.date=2004&rft.aulast=Athmaraman&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaprekar,_D._R.1974" class="citation journal cs1">Kaprekar, D. R. (1974). "The Copernicus Magic Square". <i>Indian Journal of History of Science</i>. <b>9</b> (1).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indian+Journal+of+History+of+Science&rft.atitle=The+Copernicus+Magic+Square&rft.volume=9&rft.issue=1&rft.date=1974&rft.au=Kaprekar%2C+D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurns1975" class="citation book cs1">Burns, Marilyn (1975). <i>The I Hate Mathematics Book</i>. Boston: Little Brown and Company. p. 85. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-316-11741-2" title="Special:BookSources/0-316-11741-2"><bdi>0-316-11741-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+I+Hate+Mathematics+Book&rft.place=Boston&rft.pages=85&rft.pub=Little+Brown+and+Company&rft.date=1975&rft.isbn=0-316-11741-2&rft.aulast=Burns&rft.aufirst=Marilyn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaprekar1955" class="citation journal cs1">Kaprekar, D.R. (1955). "<span class="cs1-kern-left"></span>"An interesting property of the number 6174"<span class="cs1-kern-right"></span>". <i><a href="/wiki/Scripta_Mathematica" title="Scripta Mathematica">Scripta Mathematica</a></i>. <b>21</b>: 304 – via Elsevier Science Direct.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scripta+Mathematica&rft.atitle=%22An+interesting+property+of+the+number+6174%22&rft.volume=21&rft.pages=304&rft.date=1955&rft.aulast=Kaprekar&rft.aufirst=D.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://mathpoint.blogspot.com/2006/12/mysterious-6174-revisited.html">An informal proof of the property for three digits</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://plus.maths.org/issue38/features/nishiyama/index.html">"Mysterious number 6174"</a> in <a href="/wiki/Plus_Magazine" title="Plus Magazine">Plus Magazine</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Kaprekar_Number"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/KaprekarNumber.html">"Kaprekar Number"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Kaprekar+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FKaprekarNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span></span> </li> <li id="cite_note-devlali-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-devlali_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-devlali_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Kaprekar, D. R. The Mathematics of New Self-Numbers Devalali (1963)nn: 19–20</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGunjikar,_K._R.Kaprekar,_D._R.1939" class="citation journal cs1">Gunjikar, K. R.; Kaprekar, D. R. (1939). <a rel="nofollow" class="external text" href="http://OEIS.org/A249605/a249605.pdf">"Theory of Demlo numbers"</a> <span class="cs1-format">(PDF)</span>. <i>J. Univ. Bombay</i>. <b>VIII</b> (3): 3–9.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Univ.+Bombay&rft.atitle=Theory+of+Demlo+numbers&rft.volume=VIII&rft.issue=3&rft.pages=3-9&rft.date=1939&rft.au=Gunjikar%2C+K.+R.&rft.au=Kaprekar%2C+D.+R.&rft_id=http%3A%2F%2FOEIS.org%2FA249605%2Fa249605.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Demlo_Number"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DemloNumber.html">"Demlo Number"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Demlo+Number&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDemloNumber.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AD.+R.+Kaprekar" class="Z3988"></span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=D._R._Kaprekar&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://plus.maths.org/issue38/features/nishiyama/index.html">"Mysterious number 6174"</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=d8TRcZklX_Q">Numberphile (Dec 5, 2011) 6174</a> a YouTube video by <a href="/wiki/Numberphile" title="Numberphile">Numberphile</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q560998#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q560998#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q560998#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://viaf.org/viaf/43043421">VIAF</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/133997138">Germany</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/nb2014026763">United States</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Academics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://zbmath.org/authors/?q=ai:kaprekar.d-r">zbMATH</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/MRAuthorID/262383">MathSciNet</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.deutsche-biographie.de/pnd133997138.html?language=en">Deutsche Biographie</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=D._R._Kaprekar&oldid=1256679320">https://en.wikipedia.org/w/index.php?title=D._R._Kaprekar&oldid=1256679320</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:1905_births" title="Category:1905 births">1905 births</a></li><li><a href="/wiki/Category:1986_deaths" title="Category:1986 deaths">1986 deaths</a></li><li><a href="/wiki/Category:20th-century_Indian_mathematicians" title="Category:20th-century Indian mathematicians">20th-century Indian mathematicians</a></li><li><a href="/wiki/Category:Indian_number_theorists" title="Category:Indian number theorists">Indian number theorists</a></li><li><a href="/wiki/Category:Recreational_mathematicians" title="Category:Recreational mathematicians">Recreational mathematicians</a></li><li><a href="/wiki/Category:Magic_squares" title="Category:Magic squares">Magic squares</a></li><li><a href="/wiki/Category:People_from_Thane_district" title="Category:People from Thane district">People from Thane district</a></li><li><a href="/wiki/Category:Scientists_from_Maharashtra" title="Category:Scientists from Maharashtra">Scientists from Maharashtra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_Hindi-language_sources_(hi)" title="Category:CS1 Hindi-language sources (hi)">CS1 Hindi-language sources (hi)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:EngvarB_from_September_2014" title="Category:EngvarB from September 2014">EngvarB from September 2014</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_September_2014" title="Category:Use dmy dates from September 2014">Use dmy dates from September 2014</a></li><li><a href="/wiki/Category:Articles_with_hCards" title="Category:Articles with hCards">Articles with hCards</a></li><li><a href="/wiki/Category:Articles_containing_Marathi-language_text" title="Category:Articles containing Marathi-language text">Articles containing Marathi-language text</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2024" title="Category:Articles with unsourced statements from September 2024">Articles with unsourced statements from September 2024</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2020" title="Category:Articles with unsourced statements from May 2020">Articles with unsourced statements from May 2020</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 11 November 2024, at 02:13<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=D._R._Kaprekar&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-89558","wgBackendResponseTime":171,"wgPageParseReport":{"limitreport":{"cputime":"0.601","walltime":"0.827","ppvisitednodes":{"value":3043,"limit":1000000},"postexpandincludesize":{"value":36197,"limit":2097152},"templateargumentsize":{"value":3042,"limit":2097152},"expansiondepth":{"value":17,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":43619,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 784.909 1 -total"," 22.01% 172.783 1 Template:Infobox_person"," 21.39% 167.871 1 Template:Reflist"," 17.70% 138.952 1 Template:Short_description"," 14.15% 111.050 19 Template:Main_other"," 13.59% 106.661 1 Template:Langx"," 11.04% 86.623 2 Template:Cite_web"," 10.81% 84.817 1 Template:SDcat"," 9.42% 73.970 1 Template:Authority_control"," 8.11% 63.635 4 Template:Br_separated_entries"]},"scribunto":{"limitreport-timeusage":{"value":"0.390","limit":"10.000"},"limitreport-memusage":{"value":21511220,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-4qzqr","timestamp":"20241122141016","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"D. R. Kaprekar","url":"https:\/\/en.wikipedia.org\/wiki\/D._R._Kaprekar","sameAs":"http:\/\/www.wikidata.org\/entity\/Q560998","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q560998","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-03-29T13:11:28Z","dateModified":"2024-11-11T02:13:25Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/en\/b\/b1\/D._R._Kaprekar.gif","headline":"Indian mathematician (1905\u20131986)"}</script> </body> </html>