{"id":195,"date":"2026-10-03T12:38:32","date_gmt":"2026-10-03T11:38:32","guid":{"rendered":"https:\/\/wpcalc.com\/tr\/cin-kalanlar-teoremi\/"},"modified":"2026-10-03T12:38:32","modified_gmt":"2026-10-03T11:38:32","slug":"cin-kalanlar-teoremi","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/tr\/matematik\/cin-kalanlar-teoremi\/","title":{"rendered":"\u00c7in Kalan Teoremi Hesaplay\u0131c\u0131"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">\u00c7in Kalan Teoremi \u00c7\u00f6z\u00fcc\u00fc<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"195\" data-favorite-title=\"\u00c7in Kalan Teoremi\" data-favorite-url=\"https:\/\/wpcalc.com\/tr\/matematik\/cin-kalanlar-teoremi\/\" data-favorite-excerpt=\"\u00c7in Kalan Teoremi&#039;ni kullanarak e\u015fzamanl\u0131 kongr\u00fcanslar sistemini \u00e7\u00f6zer. En k\u00fc\u00e7\u00fck negatif olmayan \u00e7\u00f6z\u00fcm\u00fc elde etmek i\u00e7in kalanlar\u0131 ve ikili aralar\u0131nda asal modlar\u0131 girin.\" aria-label=\"Favorilere ekle\" data-favorite-icon=\"icon icon-arithmetic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/tr\/matematik\/cin-kalanlar-teoremi\/\" method=\"POST\" class=\"calculator\" id=\"calculator-195\" data-post=\"195\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Giri\u015f alanlar\u0131<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Kongr\u00fcans say\u0131s\u0131<\/label><input type=\"text\" name=\"kongrans_says\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"2\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">1. kalan<\/label><div class=\"term\">a\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"1_kalan\" id=\"field-2\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">1. mod\u00fcl<\/label><div class=\"term\">n\u2081<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"1_modl\" id=\"field-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">2. kalan<\/label><div class=\"term\">a\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"2_kalan\" id=\"field-4\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">2. mod\u00fcl<\/label><div class=\"term\">n\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"2_modl\" id=\"field-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"1\" value=\"5\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Yeniden hesapla\" id=\"calculate-button\" data-post=\"195\">Hesapla<\/button><button type=\"reset\">S\u0131f\u0131rla<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Otomatik hesapla<\/label><small>Etkinle\u015ftirilirse, herhangi bir de\u011feri de\u011fi\u015ftirdi\u011finizde sonu\u00e7 otomatik olarak g\u00fcncellenir.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Sonu\u00e7lar\u0131n\u0131z<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>En k\u00fc\u00e7\u00fck negatif olmayan \u00e7\u00f6z\u00fcm<\/span><span class=\"result-term\">x<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Mod (mod\u00fcllerin EKOK&#039;u)<\/span><span class=\"result-term\">N<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Bu hesaplaman\u0131n ba\u011flant\u0131s\u0131n\u0131 kopyala<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Yorum b\u0131rak<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"195\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Bu sonu\u00e7 hakk\u0131nda ne d\u00fc\u015f\u00fcn\u00fcyorsunuz?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Be\u011fendim <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Faydal\u0131 <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Kafam kar\u0131\u015ft\u0131 <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Hayal k\u0131r\u0131kl\u0131\u011f\u0131 <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Hatal\u0131 <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>\u00c7in Kalan Teoremi Form\u00fcl\u00fc<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Form\u00fcl<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ x \\equiv a_1 \\pmod{n_1} \\\\ x \\equiv a_2 \\pmod{n_2} \\\\ \\vdots \\\\ x \\equiv a_k \\pmod{n_k} \\\\[1.5em] N = n_1 \\cdot n_2 \\cdots n_k, \\quad N_i = \\frac{N}{n_i}, \\quad y_i \\equiv N_i^{-1} \\pmod{n_i} \\\\[1.5em] x \\equiv \\sum_{i=1}^{k} a_i \\, N_i \\, y_i \\pmod{N} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Burada:<\/strong><\/p>\n<ul>\n<li>\\(a_i\\) \u2014 kalanlar (\\(x\\)'in kongr\u00fcant olmas\u0131 gereken de\u011ferler)<\/li>\n<li>\\(n_i\\) \u2014 modlar; ikili aralar\u0131nda asal olmal\u0131d\u0131rlar (herhangi ikisinin en b\u00fcy\u00fck ortak b\u00f6leni 1'dir)<\/li>\n<li>\\(N\\) \u2014 t\u00fcm modlar\u0131n \u00e7arp\u0131m\u0131; modlar aralar\u0131nda asal oldu\u011funda bunlar\u0131n en k\u00fc\u00e7\u00fck ortak kat\u0131na e\u015fittir<\/li>\n<li>\\(N_i\\) \u2014 \\(n_i\\) d\u0131\u015f\u0131ndaki t\u00fcm modlar\u0131n \u00e7arp\u0131m\u0131<\/li>\n<li>\\(y_i\\) \u2014 \\(N_i\\)'nin \\(n_i\\) moduna g\u00f6re mod\u00fcler tersi; yani \\(N_i y_i \\equiv 1 \\pmod{n_i}\\) olan say\u0131<\/li>\n<li>\\(x\\) \u2014 \\(N\\) moduna g\u00f6re tek \u00e7\u00f6z\u00fcm<\/li>\n<\/ul>\n<p>Teorem, modlar ikili aralar\u0131nda asal oldu\u011funda sistemin \\(0 \\le x &lt; N\\) aral\u0131\u011f\u0131nda tam olarak bir \u00e7\u00f6z\u00fcm\u00fc oldu\u011funu s\u00f6yler. Form\u00fcl, bu \u00e7\u00f6z\u00fcm\u00fc her bir kalan\u0131 ilgili k\u0131smi \u00e7arp\u0131m\u0131n tersiyle birle\u015ftirerek olu\u015fturur.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>\u00c7in Kalan Teoremi \u2013 Hesaplama \u00d6rne\u011fi<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>\u00d6rnek:<\/strong> \\(x \\equiv 2 \\pmod{3}\\) ve \\(x \\equiv 3 \\pmod{5}\\) ko\u015fullar\u0131n\u0131 sa\u011flayan \\(x\\)'i bulun \u2014 bunlar hesaplay\u0131c\u0131n\u0131n varsay\u0131lan de\u011ferleridir.<\/p>\n<p>\\[ N = 3 \\times 5 = 15 \\]<\/p>\n<p>\\[ N_1 = \\frac{15}{3} = 5, \\quad y_1 = 2 \\; (\\text{\u00e7\u00fcnk\u00fc } 5 \\cdot 2 = 10 \\equiv 1 \\pmod{3}) \\]<\/p>\n<p>\\[ N_2 = \\frac{15}{5} = 3, \\quad y_2 = 2 \\; (\\text{\u00e7\u00fcnk\u00fc } 3 \\cdot 2 = 6 \\equiv 1 \\pmod{5}) \\]<\/p>\n<p>\\[ x \\equiv (2 \\cdot 5 \\cdot 2) + (3 \\cdot 3 \\cdot 2) \\pmod{15} \\equiv 20 + 18 = 38 \\equiv 8 \\pmod{15} \\]<\/p>\n<p><strong>Sonu\u00e7:<\/strong> \\(x = 8\\) (mod \\(15\\)). Ger\u00e7ekten de \\(8 \\div 3 = 2\\) olup kalan \\(2\\), ve \\(8 \\div 5 = 1\\) olup kalan \\(3\\)'t\u00fcr; dolay\u0131s\u0131yla her iki kongr\u00fcans da sa\u011flan\u0131r. Bir sonraki \u00e7\u00f6z\u00fcm \\(8 + 15 = 23\\), ard\u0131ndan \\(38\\) ve b\u00f6ylece s\u00fcrer.<\/p>\n<p>\n\r\n<\/section>\n<h2>Hesaplay\u0131c\u0131 Nas\u0131l Kullan\u0131l\u0131r?<\/h2>\n<ol>\n<li><strong>Kongr\u00fcans say\u0131s\u0131n\u0131<\/strong> girin (en az 2). Yeni kalan\/mod \u00e7iftleri otomatik olarak g\u00f6r\u00fcn\u00fcr.<\/li>\n<li>\u0130lk kongr\u00fcans i\u00e7in kalan\u0131 <strong>kalan 1<\/strong> (\\(a_1\\)) alan\u0131na, modu <strong>mod 1<\/strong> (\\(n_1\\)) alan\u0131na yaz\u0131n.<\/li>\n<li>Ayn\u0131 i\u015flemi <strong>kalan 2<\/strong> \/ <strong>mod 2<\/strong> ve varsa di\u011fer \u00e7iftler i\u00e7in tekrarlay\u0131n.<\/li>\n<li><strong>En k\u00fc\u00e7\u00fck negatif olmayan \u00e7\u00f6z\u00fcm<\/strong> \\(x\\)'i ve <strong>Mod (modlar\u0131n EKOK'u)<\/strong> \\(N\\)'yi okuyun. \\(x + kN\\) (\\(k = 0, 1, 2, \\dots\\)) bi\u00e7imindeki her say\u0131 da sistemi sa\u011flar.<\/li>\n<\/ol>\n<h2>\u00c7in Kalan Teoremi Nedir?<\/h2>\n<p>\u00c7in Kalan Teoremi (\u00c7KT), say\u0131lar teorisinin klasik bir sonucudur ve ilk olarak MS 3.\u20135. y\u00fczy\u0131llar aras\u0131nda \u00c7inli matematik\u00e7i Sunzi taraf\u0131ndan tan\u0131mlanm\u0131\u015ft\u0131r. Teorem \u015fu soruyu yan\u0131tlar: birka\u00e7 b\u00f6len ile bilinmeyen bir say\u0131n\u0131n bunlara b\u00f6l\u00fcnmesinden kalan kalanlar verildi\u011finde, o say\u0131 geri bulunabilir mi? Teoremin yan\u0131t\u0131 evettir \u2014 modlar ikili aralar\u0131nda asal oldu\u011fu s\u00fcrece \u2014 ve cevap, uzunlu\u011fu t\u00fcm modlar\u0131n \u00e7arp\u0131m\u0131 olan bir aral\u0131k i\u00e7inde tektir.<\/p>\n<p>Bunu somutla\u015ft\u0131rman\u0131n bir yolu: bir yumurta sepetinde \\(x\\) yumurta varsa ve 3'erli say\u0131nca 2 art\u0131yorsa, 5'erli say\u0131nca da 3 art\u0131yorsa, \\(x \\equiv 2 \\pmod{3}\\) ve \\(x \\equiv 3 \\pmod{5}\\) olur. \u00c7KT, \\(x\\)'i mod 15'e g\u00f6re tek bi\u00e7imde belirler \u2014 bu \u00f6rnekte \\(x = 8, 23, 38, \\dots\\)<\/p>\n<p>Teoremin \u00f6nemi, bir\u00e7ok k\u00fc\u00e7\u00fck kongr\u00fcansl\u0131k sisteminin b\u00fcy\u00fck modlu tek bir kongr\u00fcansa denk oldu\u011funu g\u00f6stermesinden gelir. Bu sayede \u00e7ok b\u00fcy\u00fck bir modla yap\u0131lan aritmetik, birka\u00e7 k\u00fc\u00e7\u00fck ve kullan\u0131\u015fl\u0131 modla yap\u0131lan paralel aritmetikle de\u011fi\u015ftirilebilir ve sonu\u00e7lar sonra yeniden birle\u015ftirilebilir. Birle\u015ftirme ad\u0131m\u0131 mod\u00fcler tersleri kullan\u0131r; bunlar\u0131 <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/moduler-carpimsal-ters\/\">\u00c7arp\u0131msal Mod Tersi hesaplay\u0131c\u0131s\u0131yla<\/a> ayr\u0131ca hesaplayabilirsiniz; bu t\u00fcr tersler ayr\u0131ca <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/fermat-kucuk-teoremi\/\">Fermat'n\u0131n K\u00fc\u00e7\u00fck Teoremi<\/a> ve <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/euler-teoremi\/\">Euler Teoremi<\/a> ile de ilgilidir.<\/p>\n<h2>Sonucun Yorumlanmas\u0131<\/h2>\n<p>Hesaplay\u0131c\u0131 en k\u00fc\u00e7\u00fck negatif olmayan \u00e7\u00f6z\u00fcm \\(x\\)'i ve birle\u015fik mod \\(N\\)'yi verir. Tam \u00e7\u00f6z\u00fcm k\u00fcmesi \\(x, x + N, x + 2N, \\dots\\) bi\u00e7imindedir; probleminizde negatif cevaplar anlaml\u0131ysa \\(x - N\\) ve daha k\u00fc\u00e7\u00fck de\u011ferler de ge\u00e7erlidir. \u00c7\u00f6z\u00fcm \\(N\\) moduna g\u00f6re tek oldu\u011fundan, ayn\u0131 \\(x \\pmod{N}\\) de\u011ferini \u00fcreten iki farkl\u0131 giri\u015f tam olarak ayn\u0131 sistemi tan\u0131mlar.<\/p>\n<h2>Varsay\u0131mlar ve S\u0131n\u0131rlamalar<\/h2>\n<ul>\n<li><strong>Modlar\u0131n ikili aralar\u0131nda asal olmas\u0131 gerekir.<\/strong> Her mod \u00e7ifti 1'den b\u00fcy\u00fck ortak bir \u00e7arpan payla\u015fmamal\u0131d\u0131r; aksi h\u00e2lde hesaplay\u0131c\u0131 hata bildirir. \u00d6rne\u011fin iki mod 4 ve 6 ise (ikisi de 2 ile b\u00f6l\u00fcnebilir), sistemin hi\u00e7 \u00e7\u00f6z\u00fcm\u00fc olmayabilir ya da EKOK'lar\u0131 moduna g\u00f6re birden \u00e7ok \u00e7\u00f6z\u00fcm bulunabilir.<\/li>\n<li><strong>Sistemin tutarl\u0131l\u0131\u011f\u0131.<\/strong> Modlar aralar\u0131nda asal oldu\u011funda teorem her zaman bir \u00e7\u00f6z\u00fcm verir \u2014 \u00c7KT'nin g\u00fcc\u00fc budur. Kongr\u00fcanslar aras\u0131ndaki \u00e7eli\u015fkiler yaln\u0131zca aralar\u0131nda asall\u0131k ko\u015fulu bozuldu\u011funda ortaya \u00e7\u0131kabilir.<\/li>\n<li><strong>Kalanlar moddan b\u00fcy\u00fck olabilir.<\/strong> \\(a = 7, n = 5\\) gibi bir giri\u015f \\(7 \\equiv 2 \\pmod 5\\) olarak ele al\u0131n\u0131r; hesaplay\u0131c\u0131 nihai cevab\u0131 otomatik olarak \\(N\\) moduna g\u00f6re indirger.<\/li>\n<li><strong>Modlar pozitif tam say\u0131 olmal\u0131d\u0131r.<\/strong> 0 veya negatif de\u011ferler reddedilir.<\/li>\n<li><strong>B\u00fcy\u00fck sistemler.<\/strong> \\(N\\) t\u00fcm modlar\u0131n \u00e7arp\u0131m\u0131 oldu\u011fundan \u00e7ok h\u0131zl\u0131 b\u00fcy\u00fcr. \u00c7ok say\u0131da mod varsa, s\u00f6z konusu say\u0131lar standart kayan nokta aritmeti\u011finin tam olarak \u00e7al\u0131\u015ft\u0131\u011f\u0131 aral\u0131\u011f\u0131 a\u015fabilir.<\/li>\n<\/ul>\n<h2>\u00c7KT Nerede Kullan\u0131l\u0131r?<\/h2>\n<p>Saf say\u0131lar teorisinin \u00f6tesinde \u00c7KT, birka\u00e7 alanda etkin bir ara\u00e7t\u0131r. Kriptografide, sonucu her asal \u00e7arpan moduna ayr\u0131 ayr\u0131 hesaplay\u0131p birle\u015ftirerek RSA \u015fifre \u00e7\u00f6z\u00fcm\u00fcn\u00fc h\u0131zland\u0131r\u0131r; ayr\u0131ca Shamir'in Gizli Payla\u015f\u0131m\u0131 gibi e\u015fik \u015femalar\u0131n\u0131n temelini olu\u015fturur. Bilgisayar biliminde, k\u00fc\u00e7\u00fck modlar \u00fczerinde toplama ve \u00e7arpma i\u015flemlerini paralel olarak ger\u00e7ekle\u015ftiren rezid\u00fc (kalan) say\u0131 sistemlerini olanakl\u0131 k\u0131lar; ayr\u0131ca hashing, \u00e7izelgeleme ve takvim hesaplar\u0131nda kar\u015f\u0131m\u0131za \u00e7\u0131kar \u2014 Sunzi'nin \u00e7al\u0131\u015fmalar\u0131na ilk ilham\u0131 veren klasik \"askerleri s\u0131ralara g\u00f6re sayma\" problemleri.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00c7in Kalan Teoremi'ni kullanarak e\u015fzamanl\u0131 kongr\u00fcanslar sistemini \u00e7\u00f6zer. En k\u00fc\u00e7\u00fck negatif olmayan \u00e7\u00f6z\u00fcm\u00fc elde etmek i\u00e7in kalanlar\u0131 ve ikili aralar\u0131nda asal modlar\u0131 girin.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[18],"class_list":["post-195","post","type-post","status-publish","format-standard","hentry","category-matematik","tag-aritmetik-hesap-makineleri"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/195","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/comments?post=195"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/195\/revisions"}],"predecessor-version":[{"id":1886,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/195\/revisions\/1886"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/media?parent=195"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/categories?post=195"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/tags?post=195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}