{"id":4021,"date":"2026-10-14T23:42:54","date_gmt":"2026-10-14T22:42:54","guid":{"rendered":"https:\/\/wpcalc.com\/tr\/matematik\/moduler-us-alma-hesaplama\/"},"modified":"2026-10-14T23:42:54","modified_gmt":"2026-10-14T22:42:54","slug":"moduler-us-alma-hesaplama","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/tr\/matematik\/moduler-us-alma-hesaplama\/","title":{"rendered":"Mod\u00fcler \u00dcs Alma Hesaplama Arac\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\">Kuvvetin m mod\u00fcl\u00fcne g\u00f6re sonucunu hesaplay\u0131n<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"4021\" data-favorite-title=\"Mod\u00fcler \u00dcs Alma\" data-favorite-url=\"https:\/\/wpcalc.com\/tr\/matematik\/moduler-us-alma-hesaplama\/\" data-favorite-excerpt=\"b^e mod m de\u011ferini, \u00fcs y\u00fczlerce basamakl\u0131 olsa bile, h\u0131zl\u0131 kare al\u2013\u00e7arp y\u00f6ntemiyle an\u0131nda hesaplar. Bir taban, negatif olmayan bir \u00fcs ve pozitif bir mod girin; isterseniz t\u00fcm kare alma ad\u0131mlar\u0131n\u0131 da g\u00f6r\u00fcn.\" 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\/moduler-us-alma-hesaplama\/\" method=\"POST\" class=\"calculator\" id=\"calculator-4021\" data-post=\"4021\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Giri\u015f alanlar\u0131<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Taban (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"taban_b\" id=\"field-1\" aria-describedby=\"calc-info-1\" placeholder=\"\u00f6rn. 3 veya -12\" step=\"1\" value=\"3\"\/><small id=\"calc-info-1\">Pozitif veya negatif herhangi bir tam say\u0131<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">\u00dcs (e)<\/label><div class=\"term\">e<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"s_e\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 644\" min=\"0\" value=\"644\"\/><small id=\"calc-info-2\">Negatif olmayan tam say\u0131; b\u1d49&#039;nin do\u011frudan hesaplanamayacak kadar b\u00fcy\u00fck olmas\u0131na gerek yoktur<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Mod\u00fcl (m)<\/label><div class=\"term\">m<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"modl_m\" id=\"field-3\" aria-describedby=\"calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 645\" min=\"1\" value=\"645\"\/><small id=\"calc-info-3\">Pozitif tam say\u0131<\/small><\/div><div class=\"field is-checkbox\" id=\"input-4\"><input type=\"checkbox\" id=\"field-4\" value=\"1\" checked=\"checked\"\/><label for=\"field-4\">Kareleme ad\u0131mlar\u0131n\u0131 g\u00f6ster<\/label><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Yeniden hesapla\" id=\"calculate-button\" data-post=\"4021\">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>Sonu\u00e7 (b\u1d49 mod m)<\/span><span class=\"result-term\">b\u1d49 mod m<\/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>\u0130ndirgenmi\u015f taban (b mod m)<\/span><span class=\"result-term\">b mod m<\/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 class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Kareleme ad\u0131mlar\u0131<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/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=\"4021\" 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>Mod\u00fcler \u00dcs Alma 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\\[ c \\;=\\; b^e \\bmod m \\]\n\\[ e = \\sum_{k=0}^{n} e_k\\,2^k \\;\\;\\Longrightarrow\\;\\; b^e \\;\\equiv\\; \\prod_{k:\\; e_k = 1} \\bigl(b^{2^k} \\bmod m\\bigr) \\pmod{m}, \\qquad b^{2^k} \\equiv \\bigl(b^{2^{k-1}}\\bigr)^2 \\pmod{m} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Burada:<\/strong><\/p>\n<ul>\n<li>\\(b\\) \u2014 taban (herhangi bir tam say\u0131)<\/li>\n<li>\\(e\\) \u2014 \u00fcs (negatif olmayan bir tam say\u0131)<\/li>\n<li>\\(m\\) \u2014 mod (pozitif bir tam say\u0131)<\/li>\n<li>\\(e_k\\) \u2014 \\(e\\) say\u0131s\u0131n\u0131n ikili g\u00f6sterimindeki bitler<\/li>\n<li>\\(c\\) \u2014 sonu\u00e7: \\(b^e\\) say\u0131s\u0131n\u0131n \\(m\\) say\u0131s\u0131na b\u00f6l\u00fcnmesinden kalan<\/li>\n<\/ul>\n<p>\u0130lk sat\u0131r mod\u00fcler \u00fcs alma i\u015flemini tan\u0131mlar. \u0130kinci sat\u0131r ise hesaplama arac\u0131n\u0131n kulland\u0131\u011f\u0131 kare al\u2013\u00e7arp (ikili \u00fcs alma) \u00f6zde\u015fli\u011fidir: \u00fcs ikinin kuvvetlerine ayr\u0131\u015ft\u0131r\u0131l\u0131r, \\(b^{2^k}\\) kuvvetleri art arda kare al\u0131narak bulunur ve yaln\u0131zca 1 olan bitlere kar\u015f\u0131l\u0131k gelenler \u00e7arp\u0131l\u0131r \u2014 her \u015fey mod \\(m\\) \u00fczerinden yap\u0131ld\u0131\u011f\u0131 i\u00e7in ara de\u011ferler hi\u00e7bir zaman kontrolden \u00e7\u0131kmaz.<\/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>Mod\u00fcler \u00dcs Alma \u2013 \u00d6rnek Hesaplama<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>\u00d6rnek:<\/strong> hesaplama arac\u0131n\u0131n varsay\u0131lan de\u011ferleri olan \\(3^{644} \\bmod 645\\) i\u015flemini hesaplayal\u0131m. \\(3^{644}\\) say\u0131s\u0131 308 basamakl\u0131 oldu\u011fundan, \u00f6nce kuvveti hesaplay\u0131p sonra kalan\u0131 almaya \u00e7al\u0131\u015fmak nafile bir \u00e7abad\u0131r.<\/p>\n<p><strong>1. Ad\u0131m \u2014 taban\u0131 indirgeyin:<\/strong> \\(3 \\bmod 645 = 3\\).<\/p>\n<p><strong>2. Ad\u0131m \u2014 \u00fcss\u00fc ikilik sistemde yaz\u0131n:<\/strong><\/p>\n<p>\\[ 644 = 512 + 128 + 4 = 1010000100_2 \\]<\/p>\n<p>Mod 645 \u00fczerinde art arda kare alma, 1 olan bitler i\u00e7in \\(3^4 \\equiv 81\\), \\(3^{128} \\equiv 396\\) ve \\(3^{512} \\equiv 111 \\pmod{645}\\) de\u011ferlerini verir.<\/p>\n<p><strong>3. Ad\u0131m \u2014 bu de\u011ferleri her ad\u0131mda mod 645'e indirgeyerek \u00e7arp\u0131n:<\/strong><\/p>\n<p>\\[ 3^{644} \\equiv 81 \\times 396 \\times 111 \\equiv 471 \\times 111 \\equiv 52281 \\equiv 36 \\pmod{645} \\]<\/p>\n<p><strong>Sonu\u00e7:<\/strong> \\(3^{644} \\bmod 645 = 36\\). Algoritma yaln\u0131zca 9 kare alma ve 3 \u00e7arpma, yani basit bir y\u00f6ntemin gerektirece\u011fi 643 i\u015flem yerine toplam 12 mod\u00fcler i\u015flemle yetindi.<\/p>\n<p>\n\r\n<\/section>\n<h2>Hesaplama arac\u0131 nas\u0131l kullan\u0131l\u0131r?<\/h2>\n<ol>\n<li><strong>Taban (b)<\/strong> de\u011ferini girin \u2014 pozitif veya negatif herhangi bir tam say\u0131 olabilir.<\/li>\n<li><strong>\u00dcs (e)<\/strong> de\u011ferini girin \u2014 negatif olmayan bir tam say\u0131. \\(b^e\\)'nin do\u011frudan hesaplanamayaca\u011f\u0131 kadar b\u00fcy\u00fck bile olabilir; bu ara\u00e7 tam olarak bunun i\u00e7indir.<\/li>\n<li><strong>Mod (m)<\/strong> de\u011ferini girin \u2014 herhangi bir pozitif tam say\u0131.<\/li>\n<li>\u00dcss\u00fcn ikili a\u00e7\u0131l\u0131m\u0131n\u0131 ve her kare alma ad\u0131m\u0131n\u0131 g\u00f6steren tabloyu g\u00f6rmek istiyorsan\u0131z <strong>Kare alma ad\u0131mlar\u0131n\u0131 g\u00f6ster<\/strong> kutucu\u011funu i\u015faretli b\u0131rak\u0131n; yaln\u0131zca d\u00fcz bir sonu\u00e7 istiyorsan\u0131z i\u015fareti kald\u0131r\u0131n.<\/li>\n<li><strong>Sonu\u00e7 (b\u1d49 mod m)<\/strong> de\u011ferini, <strong>\u0130ndirgenmi\u015f taban\u0131 (b mod m)<\/strong> ve ad\u0131m ad\u0131m \u00e7\u00f6z\u00fcm\u00fc okuyun.<\/li>\n<\/ol>\n<h2>Mod\u00fcler \u00fcs alma nedir?<\/h2>\n<p>Mod\u00fcler \u00fcs alma \u015fu soruya cevap verir: <em>\\(b^e\\) say\u0131s\u0131 \\(m\\) say\u0131s\u0131na b\u00f6l\u00fcnd\u00fc\u011f\u00fcnde kalan nedir?<\/em> Sonu\u00e7 her zaman \\(0\\) ile \\(m-1\\) aras\u0131ndad\u0131r. Ba\u015fka bir deyi\u015fle bu, \\(0 \\le c &lt; m\\) olan ve \\(b^e \\equiv c \\pmod{m}\\) e\u015fli\u011fini sa\u011flayan \\(c\\) say\u0131s\u0131d\u0131r. \u00d6rne\u011fin \\(3^{644} \\bmod 645 = 36\\) olmas\u0131, \\(3^{644}\\) say\u0131s\u0131n\u0131n 645'e b\u00f6l\u00fcnmesinden 36 kald\u0131\u011f\u0131 anlam\u0131na gelir.<\/p>\n<p>Basit yol \u2014 \u00f6nce \\(b^e\\)'yi hesaplay\u0131p sonra kalan\u0131 almak \u2014 \u00e7ok \u00e7abuk t\u0131kan\u0131r. \\(3^{644}\\) bile 308 basamakl\u0131d\u0131r; kriptografik uygulamalarda ise y\u00fczlerce basamakl\u0131 \u00fcsler ola\u011fand\u0131r ve b\u00f6yle durumlarda \\(b^e\\), evrendeki atom say\u0131s\u0131ndan daha fazla basama\u011fa sahip olur. \u00c7\u00f6z\u00fcm kare al\u2013\u00e7arp y\u00f6ntemidir: \u00fcss\u00fc ikilik sistemde yaz\u0131n, \\(b^{2^k} \\bmod m\\) de\u011ferlerini art arda kare al\u0131p indirgeyerek elde edin, ard\u0131ndan yaln\u0131zca 1 olan bitlere kar\u015f\u0131l\u0131k gelen de\u011ferleri \u00e7arp\u0131n. Bu y\u00f6ntem \\(e-1\\) yerine yaln\u0131zca \\(O(\\log e)\\) mod\u00fcler \u00e7arpmaya ihtiya\u00e7 duyar ve her ara sonu\u00e7 mod \\(m\\) ile indirgendi\u011finden hi\u00e7bir say\u0131 yakla\u015f\u0131k \\(m^2\\) de\u011ferini a\u015fmaz.<\/p>\n<p>Mod\u00fcler \u00fcs alma, a\u00e7\u0131k anahtarl\u0131 kriptografinin bel kemi\u011fidir. RSA ile \u015fifreleme ve \u015fifre \u00e7\u00f6zme, devasa bir modlu ger\u00e7ekle\u015ftirilen tek bir mod\u00fcler \u00fcs alma i\u015flemidir; Diffie\u2013Hellman anahtar de\u011fi\u015fimi ve ElGamal \u015femas\u0131 da ona dayan\u0131r. Ayr\u0131ca Fermat ve Miller\u2013Rabin gibi asall\u0131k testlerinin de temelini olu\u015fturur; bu testler \\(a^{n-1} \\equiv 1 \\pmod n\\) olup olmad\u0131\u011f\u0131n\u0131 denetler \u2014 temeldeki say\u0131 teorisini <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> hesaplama ara\u00e7lar\u0131m\u0131zla inceleyebilirsiniz.<\/p>\n<p>Ad\u0131m g\u00f6sterimi algoritmay\u0131 \u015feffaf h\u00e2le getirir: tablo, \u00fcss\u00fcn her bir \\(k\\) bit konumunu \\(b^{2^k} \\bmod m\\) de\u011feriyle birlikte listeler; kal\u0131n sat\u0131rlar ger\u00e7ekte kullan\u0131lanlard\u0131r ve bunlar\u0131n \u00e7arp\u0131m\u0131 (mod \\(m\\) ile indirgenmi\u015f) cevab\u0131n\u0131zd\u0131r. Bilgisayar cebir sistemlerinin arka planda yapt\u0131\u011f\u0131 tam olarak budur ve bunu bir kez g\u00f6rmek, devasa \u00fcslerin neden bu kadar ucuz oldu\u011funu anlaman\u0131n en iyi yoludur.<\/p>\n<h2>Varsay\u0131mlar, s\u0131n\u0131rlar ve ipu\u00e7lar\u0131<\/h2>\n<ul>\n<li><strong>\u00dcs negatif olmamal\u0131d\u0131r.<\/strong> Negatif bir \u00fcs, \\(b\\)'nin mod\u00fcler tersini gerektirir; bu ters yaln\u0131zca \\(\\gcd(b, m) = 1\\) iken var olur. Bu durumu a\u015fmak i\u00e7in \u00f6nce tersi <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/moduler-carpimsal-ters\/\">Mod\u00fcler Ters<\/a> hesaplama arac\u0131m\u0131zla bulun, sonra \u00e7\u0131kt\u0131s\u0131n\u0131 taban olarak al\u0131p pozitif bir \u00fcsle kullan\u0131n.<\/li>\n<li><strong>Mod pozitif olmal\u0131d\u0131r.<\/strong> \\(m = 1\\) ise her tam say\u0131 0'a e\u015fliklidir, dolay\u0131s\u0131yla sonu\u00e7 0 \u00e7\u0131kar \u2014 bu bir hata de\u011fil, matematiksel olarak do\u011frudur.<\/li>\n<li><strong>Negatif taban sorun de\u011fildir.<\/strong> Taban \u00f6nce en k\u00fc\u00e7\u00fck negatif olmayan kalan\u0131na, yani \\(b \\bmod m\\) de\u011ferine indirgenir; bu y\u00fczden \\(-12\\) ile \\(m - 12\\) ayn\u0131 sonucu verir.<\/li>\n<li><strong>\u00dcs 0 oldu\u011funda sonu\u00e7 \\(1 \\bmod m\\)<\/strong> olur (yaln\u0131zca \\(m = 1\\) iken 0'd\u0131r); bu, \\(b^0 = 1\\) ile tutarl\u0131d\u0131r.<\/li>\n<li><strong>Boyut s\u0131n\u0131rlar\u0131:<\/strong> taban ve mod en fazla 1000'er basamakl\u0131, \u00fcs ise \\(10^{18}\\) kadar b\u00fcy\u00fck olabilir. Algoritma \u00fcsse g\u00f6re logaritmik \u00e7al\u0131\u015ft\u0131\u011f\u0131ndan, bu b\u00fcy\u00fckl\u00fckteki \u00fcsler bile an\u0131nda hesaplan\u0131r.<\/li>\n<li><strong>Sonucu ba\u015fka bir yoldan do\u011frulay\u0131n:<\/strong> bile\u015fik bir mod s\u00f6z konusuysa hesaplamay\u0131 asal kuvvet par\u00e7alar\u0131na b\u00f6l\u00fcp sonu\u00e7lar\u0131 yeniden birle\u015ftirebilirsiniz \u2014 sonu\u00e7lar\u0131 ba\u011f\u0131ms\u0131z olarak denetlemek i\u00e7in <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/cin-kalanlar-teoremi\/\">\u00c7in Kalan Teoremi<\/a> hesaplama arac\u0131m\u0131z\u0131 deneyin.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>b^e mod m de\u011ferini, \u00fcs y\u00fczlerce basamakl\u0131 olsa bile, h\u0131zl\u0131 kare al\u2013\u00e7arp y\u00f6ntemiyle an\u0131nda hesaplar. Bir taban, negatif olmayan bir \u00fcs ve pozitif bir mod girin; isterseniz t\u00fcm kare alma ad\u0131mlar\u0131n\u0131 da g\u00f6r\u00fcn.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[18],"class_list":["post-4021","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\/4021","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=4021"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/4021\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/media?parent=4021"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/categories?post=4021"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/tags?post=4021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}