{"id":36010,"date":"2026-01-10T18:40:42","date_gmt":"2026-01-10T18:40:42","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=36010"},"modified":"2026-01-10T18:40:42","modified_gmt":"2026-01-10T18:40:42","slug":"algorithmus-divisionis-euclideae","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/algorithmus-divisionis-euclideae\/","title":{"rendered":"Algorithmus Divisionis Euclideae"},"content":{"rendered":"<style>\np, ul, ol{\n  text-align: justify;\n}\nh1{\n  text-align:center;\n  text-transform: uppercase;\n}\nh2{\n  text-align:center;\n  text-transform: uppercase;\n  font-size:24pt;\n}\nh3{\n  text-align: center;\n  text-transform: uppercase;\n  font-size: 24px !important;\n}\n<\/style>\n<h1>Algorithmus Divisionis<\/h1>\n<p style=\"text-align:center;\"><em>In hac lectione evolvemus <strong>algorithmum divisionis<\/strong> ut principium quod, pro numeris integris, decompositionem unicam formalizat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>. Primum demonstratur <strong>existentia<\/strong> quocientis et reliqui, deinde eorum <strong>unicitas<\/strong>. Denique interpretatur significatio reliqui, coniungitur theoria cum <strong>algorithmo longo divisionis<\/strong> ut ratione calculi, atque praevidetur eius naturalis connexio cum arithmetica modulari et applicationibus computatoriis.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi<\/strong><\/p>\n<p>Hac lectione confecta, discipulus erit idoneus ad:<\/p>\n<ol>\n<li><strong>Identificare<\/strong> notiones et partes fundamentales divisionis integrae (dividendus, divisor, quociens, reliquum) atque notionem divisibilitatis ut casum exactum.<\/li>\n<li><strong>Explicare<\/strong> enuntiationem algorithmi\/theorematis divisionis Euclideae, inclusis condicionibus quae reliquum determinant (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>) et earum fine ad ambiguitates vitandas.<\/li>\n<li><strong>Applicare<\/strong> decompositionem Euclideam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> ad determinanda <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> in exemplis concretis, terminum reliqui comprobans.<\/li>\n<li><strong>Analyzare<\/strong> tractationem casuum secundum signum dividendi et divisoris, iustificans cur conventio Euclidea reliquum non negativum servet.<\/li>\n<li><strong>Exsequi<\/strong> algorithmum longum divisionis ut processum mechanicum in repraesentatione positionali fundatum ad obtinenda <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span>.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>INDEX RERUM<\/u>:<\/b><br \/>\n<a href=\"#1\">Divisio et divisibilitas<\/a><br \/>\n<a href=\"#2\">Theorema divisionis Euclideae<\/a><br \/>\n<a href=\"#3\">Demonstratio divisionis Euclideae<\/a><br \/>\n<a href=\"#4\">Interpretatio quocientis et reliqui<\/a><br \/>\n<a href=\"#5\">Algorithmus longus divisionis<\/a><br \/>\n<a href=\"#6\">Conclusio<\/a><br \/>\n<a href=\"#7\">Exercitia proposita et soluta<\/a>\n<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Divisio et divisibilitas<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/nlER6VzrGwA?si=79gpsAzwc7wcWEVn\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>In arithmetica, divisibilitas casum \u00abexactum\u00bb describit: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> significat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> exacte scribi ut multiplex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. Attamen, quamquam non semper est divisibilitas inter duos integros quoscumque, tamen quaerere possumus \u00abquoties unus numerus intra alterum conveniat et, si non exacte conveniat, quid supersit\u00bb. Hic est contextus in quo oritur <strong>algorithmus divisionis<\/strong>, qui praestat ut integer quilibet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> scribi possit ut multiplex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> addito \u00abreliquo\u00bb.<\/p>\n<p>Exempli gratia, consideremus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span>. Manifestum est 3 non dividere 16. Tamen possibile est invenire quocientem et reliquum pro operatione divisionis, quoniam<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">16 = 5\\cdot 3 + 1<\/span>\n<figure id=\"attachment_35775\" aria-describedby=\"caption-attachment-35775\" style=\"width: 1492px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg\" alt=\"Un grupo de 16 cajas han sido separadas en grupos de a 3, como resultado se obtienen 5 grupos de cajas y sobra 1\" width=\"1492\" height=\"230\" class=\"size-full wp-image-35775 lazyload\" \/><figcaption id=\"caption-attachment-35775\" class=\"wp-caption-text\"><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg\" alt=\"Un grupo de 16 cajas han sido separadas en grupos de a 3, como resultado se obtienen 5 grupos de cajas y sobra 1\" width=\"1492\" height=\"230\" class=\"size-full wp-image-35775 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3.jpg 1492w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-300x46.jpg 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-1024x158.jpg 1024w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/12\/division16por3-768x118.jpg 768w\" sizes=\"(max-width: 1492px) 100vw, 1492px\" \/><\/noscript> Grex sedecim capsarum in greges ternos divisus est; inde quinque greges capsarum obtinentur et una superest<\/figcaption><\/figure>\n<p>Quapropter, diviso <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=16<\/span><\/span> per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=3<\/span><\/span> (ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> est <strong>dividendus<\/strong> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> est <strong>divisor<\/strong>), quod repraesentatur per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\/3<\/span><\/span> vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\div 3<\/span><\/span>, obtinetur ut <strong>quociens<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=5<\/span><\/span> et ut <strong>reliquum<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>. In universum, <em>algorithmus divisionis<\/em> affirmat quod, pro quibuscumque numeris integris <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> (divisore) et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> (dividendo) cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, exsistunt integri unici <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> tales ut<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r,\\qquad 0\\le r&lt;|a|<\/span>\n<p>In hoc exemplo, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16=3\\cdot 5+1<\/span><\/span>, et condicio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> impletur, quoniam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 1&lt;3<\/span><\/span>. Haec formatio ambiguitates vitat, cum divisor (vel dividendus) negativus esse possit, atque praestat ut reliquum semper sit non negativum et stricte minus quam valor absolutus divisoris.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Theorema divisionis Euclideae<\/h2>\n<p>Effectus applicationis algorithmi divisionis est id quod <strong>divisio Euclidea<\/strong> appellatur, atque innititur sequenti enuntiato.<\/p>\n<p><b>Theorema:<\/b> Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. Tunc exsistunt <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> unici tales ut<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r,\\qquad 0\\le r&lt;|a|.\n\n<\/span>\n<p>Integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> <strong>quociens<\/strong> vocatur, et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> <strong>reliquum<\/strong> divisionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Demonstratio divisionis Euclideae<\/h2>\n<p>Haec demonstratio in duas partes dividitur: primum ostenditur quocientem et reliquum exsistere; deinde, dato quod exsistant, eos esse unicos.<\/p>\n<h3>Existentia<\/h3>\n<p>Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, et definiamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>, ita ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d&gt;0<\/span><\/span>. Primum probabimus casum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>: ostendemus exsistere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> tales ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>. Deinde tractabimus casum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span>. In fine, substituemus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d<\/span><\/span> pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span>, ut obtineatur forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>.<\/p>\n<p>Pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>, statuimus propositionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span>: \u00abexsistunt <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> tales ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq+r<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span>\u00bb. Per inductionem demonstrabimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> valere pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>.<\/p>\n<p><strong>Casus basalis:<\/strong> Pro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>, sumptis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=0<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span>, obtinetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=d\\cdot 0+0<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 0&lt;d<\/span><\/span>. Ergo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(0)<\/span><\/span> vera est.<\/p>\n<p><strong>Gradus inductivus:<\/strong> Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\ge 0<\/span><\/span> et supponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k)<\/span><\/span> veram esse. Tunc exsistunt <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> tales ut<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk=dq+r,\\qquad 0\\le r&lt;d.\n\n<\/span>\n<p>Addito <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1<\/span><\/span> utrique lateri obtinetur<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq+(r+1).\n\n<\/span>\n<p>Praeterea, ex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;d<\/span><\/span> sequitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1\\le d<\/span><\/span>. Quapropter tantum evenire possunt casus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span> vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span>.<\/p>\n<p><b>Casus 1:<\/b> Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1&lt;d<\/span><\/span>, definimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=r+1<\/span><\/span>. Tunc<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq&#039;+r&#039;,\\qquad 0\\le r&#039;&lt;d.\n\n<\/span>\n<p><b>Casus 2:<\/b> Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r+1=d<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq+d=d(q+1)+0<\/span><\/span>. Definimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;=q+1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;=0<\/span><\/span>, atque valet<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nk+1=dq&#039;+r&#039;,\\qquad 0\\le r&#039;&lt;d.\n\n<\/span>\n<p>In utroque casu integros <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q&#039;,r&#039;<\/span><\/span> construximus tales ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k+1=dq&#039;+r&#039;<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&#039;&lt;d<\/span><\/span>. Ergo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(k+1)<\/span><\/span> vera est. Concludimus per inductionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(b)<\/span><\/span> valere pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\ge 0<\/span><\/span>.<\/p>\n<p>Nunc consideremus casum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b&lt;0<\/span><\/span>. Tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b&gt;0<\/span><\/span>. Applicato superiore effectu ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-b<\/span><\/span>, exsistunt <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,r\\in\\mathbb{Z}<\/span><\/span> tales ut<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n-b=dq+r,\\qquad 0\\le r&lt;d.\n\n<\/span>\n<p>Multiplicando per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> obtinetur<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=-dq-r.\n\n<\/span>\n<p>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span>, satis est sumere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=0<\/span><\/span>, quo fit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>.<\/p>\n<p>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&gt;0<\/span><\/span>, definimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=-q-1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r_1=d-r<\/span><\/span>. Cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;r&lt;d<\/span><\/span>, habetur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0&lt;d-r&lt;d<\/span><\/span>, id est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>. Praeterea,<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\ndq_1+r_1=d(-q-1)+(d-r)=-dq-d+d-r=-dq-r=b.\n\n<\/span>\n<p>Quapropter, pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in\\mathbb{Z}<\/span><\/span> exsistunt <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1,r_1\\in\\mathbb{Z}<\/span><\/span> tales ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;d<\/span><\/span>.<\/p>\n<p>Denique, meminerimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=|a|<\/span><\/span>. Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=a<\/span><\/span>, et aequalitas <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1<\/span><\/span> scribitur ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=aq_1+r_1<\/span><\/span>, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r_1&lt;|a|<\/span><\/span>.<\/p>\n<p>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&lt;0<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d=-a<\/span><\/span>, et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=dq_1+r_1=(-a)q_1+r_1=a(-q_1)+r_1<\/span><\/span>. Definito <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-q_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r_1<\/span><\/span>, obtinetur<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r,\\qquad 0\\le r&lt;|a|.\n\n<\/span>\n<p>His ita demonstrata est existentia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> pro quibuscumque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>.<\/p>\n<h3>Unicitas<\/h3>\n<p>Supponatur exsistere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q,q&#039;,r,r&#039;\\in\\mathbb{Z}<\/span><\/span> tales ut<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nb=qa+r=q&#039;a+r&#039;,\\qquad 0\\le r,r&#039;&lt;|a|.\n\n<\/span>\n<p>Subtrahendo utrasque expressiones obtinetur<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\na(q-q&#039;)=r&#039;-r.\n\n<\/span>\n<p>Quoniam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r,r&#039;&lt;|a|<\/span><\/span>, valet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|r&#039;-r|&lt;|a|<\/span><\/span>. At vero membrum sinistrum est multiplex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, atque ideo etiam multiplex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span>. Unicum multiplex integrum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> cuius valor absolutus stricte minor est quam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0<\/span><\/span>; etenim, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a| m \\neq 0<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m\\in\\mathbb{Z}<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\big| |a| m \\big| \\ge |a| <\/span><\/span>. Quare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r&#039;-r=0<\/span><\/span>, id est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=r&#039;<\/span><\/span>, atque substituendo in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> deducitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=q&#039;<\/span><\/span>. Propterea decompositio unica est.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Interpretatio quocientis et reliqui<\/h2>\n<ul>\n<li style=\"text-align: justify;\">\n    <strong>Quid significet \u00abreliquum\u00bb.<\/strong> Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=b-qa<\/span><\/span> est id quod \u00absuperest\u00bb postquam a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> removetur multiplex <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">qa<\/span><\/span>. Aliis verbis, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> est residuum adaequationis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> ad reticulum multiplicium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>.\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>Cur exigatur<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>. Sine hac condicione, par <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(q,r)<\/span><\/span> unicum non esset. Nam, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span>, tunc etiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=a(q+1)+(r-a)<\/span><\/span>. Condicio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> cogit reliquum intra fenestram fixam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,2,\\dots,|a|-1\\}<\/span><\/span> consistere, atque ita infinitas \u00abre-nominationes\u00bb eiusdem numeri impedit.\n  <\/li>\n<li style=\"text-align: justify;\">\n    <strong>Quid accidat si<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>negativus est.<\/strong> Theorema sine mutationibus valet: reliquum manet non negativum et limite <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|<\/span><\/span> terminatum. Hoc interest, quoniam nonnulli linguarum programmandi truncationem ad zerum adhibent et reliqua negativa producere possunt, dum in mathematicis conventio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span> adhibetur, ut reliquum sit repraesentans \u00abnormatum\u00bb.<\/p>\n<p>    Praeterea, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a&gt;0<\/span><\/span>, quociens cum pavimento congruit:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nq=\\left\\lfloor \\frac{b}{a}\\right\\rfloor,\\qquad r=b-a\\left\\lfloor \\frac{b}{a}\\right\\rfloor.\n\n<\/span>\n<\/li>\n<\/ul>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Algorithmus longus divisionis<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ABw4QYdD6Pg?si=eJiA7P6cD7RdlYYN\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p>Ex divisione Euclidea fieri potest ut <strong>algorithmus longus divisionis<\/strong> impleatur, adhibita repraesentatione positionali numerorum. Hic algorithmus est ars calculandi quae sinit valores <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> celeriter inveniri, cum computare velimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\div a<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>.<\/p>\n<p>Ad eum describendum, prius nonnulla exempla inspiciamus quae processum et varias condiciones in cursu algorithmi ostendunt.<\/p>\n<ol>\n<li>\n<p>Supponamus nos velle computare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">57\\div 4<\/span><\/span>. Hic 57 est <strong>dividendus<\/strong> et 4 <strong>divisor<\/strong>. Ad hoc efficiendum, sequentem computationum seriem peragemus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n(1) &amp; \\color{red}57 \\div 4 =\\color{black}  \\\\ &amp; \\text{Operationem divisionis scribere.} \\\\ \\\\\n\n(2) &amp; \\color{red}5&#039;\\color{black}7 \\div 4 =  \\\\ &amp; \\text{Primum praefixum (a sinistra) dividendi separare quod}\\\\\n\n&amp;\\text{maiorem vel aequalem sit divisori, hoc in casu 5.} \\\\ \\\\\n\n(3) &amp; 5&#039;7 \\div 4 = \\color{red}1\\color{black} \\\\ &amp; \\text{Cogitare maximum numerum qui, multiplicatus per $4$,}\\\\\n\n&amp;\\text{minor vel aequalis sit quam $5$, eumque ad dextram aequalitatis scribere. Est 1.} \\\\ \\\\\n\n(4) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;7 \\div 4 &amp;= 1 \\\\ \\color{red}-|\\underline{4}\\color{black} &amp; \\\\ \\color{red}\\phantom{-|}1\\color{black} &amp;  \\end{array} \\\\ &amp; \\text{Resultatum per divisorem multiplicare et a valore selecto subtrahere.} \\\\ \\\\\n\n(5) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;\\color{red}7&#039;\\color{black} \\div 4 &amp;= 1 \\\\ -|\\underline{4} &amp; \\\\ \\phantom{-|}1\\color{red}7\\color{black} &amp;  \\end{array} \\\\ &amp; \\text{Sequentem digitum seligere et \u00abdemittere\u00bb.} \\\\ \\\\\n\n(6) &amp; \\begin{array}{ll}\\phantom{-|}5&#039;7&#039; \\div 4 &amp;= 1\\color{red}4\\color{black} \\\\ -|\\underline{4} &amp; \\\\ \\phantom{-|}17 &amp; \\\\ \\color{red}-|\\underline{16}\\color{black} &amp; \\\\ \\color{red}\\phantom{-|}1\\color{black}  \\end{array} \\\\ &amp; \\text{Seriem cum ultimo numero obtento iterare.}\n\n\\end{array}\n\n<\/span>\n<p>Algorithmus desinit cum digiti ad \u00abdemittendum\u00bb exhausti sunt, tradens ut eventum quocientem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=14<\/span><\/span> et reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=1<\/span><\/span>. Praesertim, reliquum semper erit minus quam divisor.<\/p>\n<\/li>\n<li>\n<p>Supponamus nos velle computare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">132\\div 5<\/span><\/span>. Ad hoc efficiendum, sequentem computationum seriem peragemus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(1) &amp;1&#039;32 \\div 5 = \\\\\n\n&amp; \\text{Primum praefixum dividendi separare et quaerere aliquem naturalem qui,}\\\\\n\n&amp; \\text{multiplicatus per 5, minor vel aequalis sit eo. Si id fieri non potest,}\\\\\n\n&amp; \\text{sequentem digitum adiungere donec fieri possit.} \\\\ \\\\\n\n(2) &amp;\\begin{array}{ll}\\phantom{-|}13&#039;2 \\div 5 &amp;= 2 \\\\ -|\\underline{10} &amp; \\\\ \\phantom{-|1}3 &amp; \\end{array}\\\\\n\n&amp; \\text{Cum gradus superior succedat, algorithmum more solito exsequi.} \\\\ \\\\\n\n(3) &amp;\\begin{array}{ll}\\phantom{-|}13&#039;2&#039; \\div 5 &amp;= 26 \\\\ -|\\underline{10} &amp; \\\\ \\phantom{-|1}32 &amp;  \\\\ \\phantom{\\,} -|\\underline{30} &amp; \\\\ \\phantom{-|30} 2 &amp;  \\end{array}\n\n\\end{array}<\/span>\n<\/li>\n<li>\n<p>Si dividendus plures digitos habet, processus eodem modo exsequitur. Exempli gratia, computando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">3521\\div 12<\/span><\/span> obtinetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\phantom{-|}35&#039;2&#039;1&#039; \\div 12 &amp; = 293 \\\\\n\n-|\\underline{24} &amp; \\\\\n\n\\phantom{-|}112 &amp; \\\\\n\n-|\\underline{108} &amp; \\\\\n\n\\phantom{-|10}41 &amp; \\\\\n\n\\phantom{0}-|\\underline{36} &amp; \\\\\n\n\\phantom{10-|}5 &amp; \\\\\n\n\\end{array}<\/span>\n<\/li>\n<li>\n<p>Si unus numerorum negativus est, duae formae acceptabiles ad eventum exhibendum exstant. Attamen, forma normata in theoria numerorum est illa in qua reliquum est positivum atque ideo cum divisione Euclidea congruens. Exempli gratia, si computamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span>, sequentes progressus possibiles habentur, ambo recti ut identitates:<\/p>\n<ul>\n<li>\n        <b>Cum reliquo negativo:<\/b> Hoc obtinetur signo in initio operationis remoto et in fine restituto.<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\phantom{-}59&#039;8&#039; \\div 21 &amp;= 28 \\\\\n\n-|\\underline{42} &amp; \\\\\n\n\\phantom{-|&#039;}178 &amp; \\\\\n\n\\phantom{\\,}-|\\underline{168} &amp; \\\\\n\n\\phantom{-|00}10 &amp; \\\\\n\n\\end{array}<\/span>\n<p>Ex hoc sequitur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = 21 \\times 28 + 10<\/span>\n<p>Deinde, si totam aequalitatem per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> multiplicamus, obtinetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10<\/span>\n<p>Itaque, hac repraesentatione, eventus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> est quociens <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span> et reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=-10<\/span><\/span>.<\/p>\n<p>Et si rursus totam aequalitatem per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/span> multiplicamus, obtinetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = (-21)\\times(-28) + 10<\/span>\n<p>Itaque eventus computationis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598\\div (-21)<\/span><\/span> est quociens <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-28<\/span><\/span> et reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=10<\/span><\/span>.<\/p>\n<\/li>\n<li>\n        <b>Cum reliquo positivo et cum divisione Euclidea congruente:<\/b> Ex superiore progressu habetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10<\/span>\n<p>Deinde, addito <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=21-21<\/span><\/span> in latere dextro huius aequalitatis, obtinetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598 = 21 \\times (-28) - 10 \\color{red}+ 21 - 21\\color{black} = 21\\times(-29) + 11<\/span>\n<p>Quapropter, divisioni Euclideae consentientes, eventus computationis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-598\\div 21<\/span><\/span> est quociens <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=-29<\/span><\/span> et reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=11<\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<p>Quoad id quod est conventionale in divisione Euclidea, solus eventus cum reliquo positivo ad quocientem et reliquum Euclidea pertinet. Attamen, utraque expressio identitates rectae sunt atque in diversis contextibus utiles esse possunt. Quamvis algorithmus longus divisionis reliqua negativa producere possit, hoc utile est ad eventus mechanice obtinendos et ad manipulationes algebraicas sine gradibus additicis perficiendas. Ex altera parte, eventus cum reliquo positivo, divisioni Euclideae congruentes, permittunt repraesentantem canonicum cuiusque classis residualis (exempli gratia in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0,1,\\dots,n-1\\}<\/span><\/span>) statui, quo integeres modo accurato et ab ambiguitate libero designentur. Notandum est etiam alias repraesentantium conventiones exsistere, ut residua symmetrica, quae aeque validae sunt, semel regula constituta.<\/p>\n<\/li>\n<\/ol>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Conclusio<\/h2>\n<p>Algorithmus divisionis praestat ut omnis divisio integra unico modo exprimi possit ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=qa+r<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r&lt;|a|<\/span><\/span>, quo criterium normatum ad quocientem et reliquum interpretandos statuitur, etiam cum numeri negativi intersint. Algorithmus longus divisionis nihil aliud est nisi implementatio practica huius decompositionis, in repraesentatione positionali innixa, quae sinit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> mechanice computari. Denique, haec forma Euclidea non solum ambiguitates vitat, sed etiam naturaliter cum arithmetica modulari coniungitur: reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> agit ut repraesentans canonicus classis residualis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> modulo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, quod fundamentum est multarum technicarum in theoria numerorum et in applicationibus computatoriis, quae in proximis expositionibus tractabuntur.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Exercitia proposita et soluta<\/h2>\n<h3>Divisio Euclidea<\/h3>\n<ol>\n<li><strong>(solutum)<\/strong> Inveni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> tales ut\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137 = 9q + r,\\qquad 0\\le r\\lt 9.<\/span>\n<p><strong>Solutio:<\/strong> Quaere numerum maximum qui, multiplicatus per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">9<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137<\/span><\/span> non superet. Habemus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">15\\times 9 = 135<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">16\\times 9 = 144<\/span><\/span>; igitur quociens quaesitus est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=15<\/span><\/span>. Nunc computa reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=137-9q<\/span><\/span> et verifica <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 9<\/span><\/span>. Verificatione facta habetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr=137-9\\cdot 15=137-135=2.\n\n<\/span>\n<p>Igitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">137=9\\cdot 15+2<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 2\\lt 9<\/span><\/span>.<\/p>\n<\/li>\n<li><strong>(solutum)<\/strong> Inveni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> tales ut\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025 = 37q + r,\\qquad 0\\le r\\lt 37.<\/span>\n<p><strong>Solutio:<\/strong> Quaere numerum maximum qui, multiplicatus per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025<\/span><\/span> non superet. Observa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 54 = 1998<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">37\\times 55 = 2035<\/span><\/span>; igitur quociens quaesitus est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q=54<\/span><\/span>. Nunc computa reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=2025-37q<\/span><\/span> et verifica <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le r\\lt 37<\/span><\/span>. Verificatione facta habetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nr=2025-37\\cdot 54=2025-1998=27.\n\n<\/span>\n<p>Igitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2025=37\\cdot 54+27<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\le 27\\lt 37<\/span><\/span>.<\/p>\n<\/li>\n<li>Inveni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> tales ut\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">745 = 23q + r,\\qquad 0\\le r\\lt 23.<\/span>\n<\/li>\n<li>Inveni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> tales ut\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-314 = 11q + r,\\qquad 0\\le r\\lt 11.<\/span>\n<\/li>\n<li>Inveni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> tales ut\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">598 = (-21)q + r,\\qquad 0\\le r\\lt |-21|.<\/span>\n<\/li>\n<li><strong>Reliquum fixum et inquisitio integrorum.<\/strong> Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=12<\/span><\/span>.\n<p>(a) Describe collectionem omnium integrorum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> quorum divisio Euclidea per <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">12<\/span><\/span> reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=5<\/span><\/span> habet.<\/p>\n<p>(b) Inveni minimum integrum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\gt 1000<\/span><\/span> quod hoc impleat, atque determina quocientem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> correspondentem.<\/p>\n<\/li>\n<\/ol>\n<h3>Algorithmus longus divisionis<\/h3>\n<p>In singulis casibus, algorithmum longum divisionis adhibe ut quocientem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q<\/span><\/span> et reliquum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r<\/span><\/span> computetur. Eventum in divisionem Euclideam converte, cum algorithmus reliquum negativum praebeat.\n<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">84\\div 6.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">197\\div 8.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1256\\div 7.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-3521\\div 12.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-98765\\div 24.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">845\\div -13.<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-12345\\div -37.<\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Algorithmus Divisionis In hac lectione evolvemus algorithmum divisionis ut principium quod, pro numeris integris, decompositionem unicam formalizat cum . Primum demonstratur existentia quocientis et reliqui, deinde eorum unicitas. Denique interpretatur significatio reliqui, coniungitur theoria cum algorithmo longo divisionis ut ratione calculi, atque praevidetur eius naturalis connexio cum arithmetica modulari et applicationibus computatoriis. Proposita Discendi Hac [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35980,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1298,1416],"tags":[],"class_list":["post-36010","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematica","category-theoria-numerorum"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Algorithmus Divisionis Euclideae - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce algorithmum divisionis et divisionem Euclideam: existentiam et unicitatem q et r, casus cum numeris negativis, algorithmum longum, exempla et exercitationes.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/algorithmus-divisionis-euclideae\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Algorithmus Divisionis Euclideae\" \/>\n<meta property=\"og:description\" content=\"Disce algorithmum divisionis et divisionem Euclideam: existentiam et unicitatem q et r, casus cum numeris negativis, algorithmum longum, exempla et exercitationes.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/algorithmus-divisionis-euclideae\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2026-01-10T18:40:42+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2026\/01\/divisioneuclidiana-1024x683.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Algorithmus Divisionis Euclideae\" \/>\n<meta name=\"twitter:description\" content=\"Disce algorithmum divisionis et divisionem Euclideam: existentiam et unicitatem q et r, casus cum numeris negativis, algorithmum longum, exempla et exercitationes.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2026\/01\/divisioneuclidiana.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Algorithmus Divisionis Euclideae\",\"datePublished\":\"2026-01-10T18:40:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/\"},\"wordCount\":2664,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/divisioneuclidiana.jpg\",\"articleSection\":[\"Mathematica\",\"Theoria Numerorum\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/algorithmus-divisionis-euclideae\\\/\",\"name\":\"Algorithmus Divisionis Euclideae - 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