{"id":35691,"date":"2025-12-26T18:56:30","date_gmt":"2025-12-26T18:56:30","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35691"},"modified":"2025-12-26T18:56:30","modified_gmt":"2025-12-26T18:56:30","slug":"quid-est-divisibilitas","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/quid-est-divisibilitas\/","title":{"rendered":"Quid est divisibilitas?"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Divisibilitas<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\n<strong>Divisibilitas<\/strong> est verum initium theoriae numerorum, quia numeros integros in systema structura praeditum convertit: iam non spectas numeros ut \u201cquantitates\u201d, sed ut partes quae inter se congruunt aut non congruunt. Una sola structura, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, sinit te exprimere tam criteria simplificationis et factorizationis quam ipsum cor quorumdam processuum, ut algorithmus Euclidis, qui efficit ut maximi communes divisores etiam numerorum magnorum intra paucos gradus computari possint. Praeterea, haec est basis technica notionum quae iterum atque iterum apparent in mathematica applicata et computatione: congruentiae, arithmetica modularis, validationes, codices, et (postea) cryptographia. Divisibilitatem dominari est, essentia tenus, discere rationes occultas in numeris integris deprehendere atque eas in processus convertere qui semper operantur.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>Proposita Discendi<\/b><br \/>\nAd finem huius apuncti discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intellegere<\/strong> relationem divisibilitatis inter numeros integros.<\/li>\n<li><strong>Intellegere<\/strong> definitionem divisibilitatis et eius proprietates.<\/li>\n<li><strong>Elaborare<\/strong> demonstrationes mathematicas de effectibus et theorematibus ad divisibilitatem pertinentibus.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>INDEX CONTENTORUM<\/u><\/b><br \/>\n<a href=\"#1\">Definitio divisibilitatis<\/a><br \/>\n<a href=\"#2\">Proprietates fundamentales divisibilitatis<\/a><br \/>\n<a href=\"#3\">Exercitationes Propositae<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/HLrwdLse18U?si=tDiiV02P7ppdb4xF\" 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><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/br><\/p>\n<h2>Definitio divisibilitatis<\/h2>\n<p style=\"text-align: justify;\">\nNotio informalis \u201c<em>a dividit b<\/em>\u201d fit precisa cum eam ut relationem inter numeros integros exprimimus. Dicemus numerum integrum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>dividere<\/strong> numerum integrum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> scribi potest ut multiplex exactum ipsius <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. Haec definitio est fundamentum reliqui apuncti, quia locutiones huius modi \u201cexacte convenit\u201d in criterium verificabile convertit.\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>Definitio.<\/strong> 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>. Dicimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> dividere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, et scribimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, si et solum si existit numerus integer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> talis ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>. Aliter, scribimus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\nmid b<\/span><\/span>.\n<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b := (\\exists k \\in \\mathbb{Z})(b = ka )<\/span>\n<p style=\"text-align: justify;\">\nIn hac definitione, numerus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> vocatur <strong>quotiens<\/strong> (vel factor) divisibilitati associatus. Exempli gratia, affirmare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> aequivalet affirmare existere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tale ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>; hoc in casu, satis est sumere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>.\n<\/p>\n<h3>Magni momenti est considerare<\/h3>\n<ul>\n<li>\n    Conditio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> est essentialis. Nam si conaremur admittere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>, condicio divisibilitatis postularet existere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tale ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>. Sed <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>, ita ut sola possibilitas esset <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>. In eo casu, nullus esset <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> a relatione \u201cdeterminatus\u201d, quippe cum quilibet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> conditionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span> satisfaciat. Aliter dictum, expressio informalis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> fit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>, quae definita non est. Ad hanc degenerationem vitandam (ubi notio quotiens significationem amittit), requiritur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. Hac de causa relatio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> valida non habetur.\n<\/li>\n<li>\n        Contra, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> verum est pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, quia satis est sumere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span> et impletur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=a\\cdot 0<\/span><\/span>.\n    <\/li>\n<\/ul>\n<p style=\"text-align: justify;\">\nEx hac definitione sequitur aequivalentia qua frequenter utemur: dicere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> idem est ac dicere <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> ad pertinere ad multitudinem multiplorum integrorum ipsius <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, id est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>, ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>. Haec scribendi ratio extollit divisibilitatem non esse \u201cartificium\u201d, sed modum describendi subcollectiones valde structas intra <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}<\/span><\/span>.<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>Proprietates fundamentales divisibilitatis<\/h2>\n<ul>\n<li><strong>Reflexivitas:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>Demonstratio<\/u>:<\/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;\\vdash a=ka \\leftrightarrow k=1 &amp;\\text{; Neutrum multiplicativum in $\\mathbb{Z}$}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; Introductio existentialis (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; Definitio divisibilitatis (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>Transitivitas:<\/strong> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>.\n<p><u>Demonstratio<\/u>:<\/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; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1\\in\\mathbb{Z})(b=k_1a)  &amp;\\text{; Def. divisibilitatis, Praesumptio}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; Def. divisibilitatis, Praesumptio}\\\\\n\n(3)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})(b=k_1a \\wedge c=k_2b)  &amp;\\text{; $\\exists$-Compactatio(1,2)}\\\\\n\n(4)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})(k_2b=k_1k_2a \\wedge c=k_2b)  &amp;\\text{; Ex(3)}\\\\\n\n(5)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_1,k_2\\in\\mathbb{Z})( c=k_1k_2a)  &amp;\\text{; Ex(4)}\\\\\n\n&amp;\\text{Algebra intra quantificatorem}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; Ex(5)}\\\\\n\n&amp;\\text{Clausura $\\mathbb{Z}$ sub multiplicatione}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; Def. divisibilitatis (6)}\\\\\n\n(8)&amp; \\vdash (a\\mid b \\wedge  b\\mid c) \\rightarrow  a\\mid c  &amp;\\text{; $\\wedge$-TD(7)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>Compatibilitas cum additione et subtractione:<\/strong> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>.<br \/>\n<u>Demonstratio<\/u>:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1 \\in \\mathbb{Z})(b=k_1 a) &amp;\\text{; Def. divisibilitatis, Praesumptio}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; Def. divisibilitatis, Praesumptio}\\\\\n\n(3)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b=k_1 a \\wedge c=k_2 a) &amp;\\text{; $\\exists$-Compactatio(1,2)}\\\\\n\n(4)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b+c= (k_1+k_2)a) &amp;\\text{; Ex(3)}\\\\\n\n&amp;\\text{Algebra intra quantificatorem.}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; Ex(4)}\\\\\n\n&amp;\\text{Clausura $\\mathbb{Z}$ sub additione.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; Def. divisibilitatis (5)}\\\\\n\n(7)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b+c) &amp;\\text{; $\\wedge$-TD(6)}\\\\\n\n(8)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_1, k_2 \\in \\mathbb{Z})(b-c= (k_1-k_2)a) &amp;\\text{; Ex(3)}\\\\\n\n&amp;\\text{Algebra intra quantificatorem.}&amp; \\\\\n\n(9)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists \\overline{k} \\in \\mathbb{Z})(b-c= \\overline{k}a) &amp;\\text{; Ex(8)}\\\\\n\n&amp;\\text{Clausura $\\mathbb{Z}$ sub subtractione.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; Def. divisibilitatis (9)}\\\\\n\n(11)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow a\\mid (b-c) &amp;\\text{; $\\wedge$-TD(10)}\\\\\n\n(12)&amp;\\vdash (a\\mid b \\wedge a\\mid c) \\rightarrow \\left(a\\mid (b+c) \\wedge a\\mid (b-c)\\right) &amp;\\text{;$\\wedge$-Introductio in consequenti(7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>Compatibilitas cum productis:<\/strong> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span>.<br \/>\n<u>Demonstratio<\/u>:<\/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; \\{a\\mid b\\}\\vdash (\\exists k\\in\\mathbb{Z})(b=ka) &amp;\\text{; Def. divisibilitatis, Praesumptio}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; Ex(1), $\\forall$-Introductio (c arbitrarium)}\\\\\n\n&amp;\\text{Algebra in }\\mathbb{Z}\\text{ intra quantificatorem existentiel.}&amp;\\\\\n\n(3)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists \\overline{k}\\in\\mathbb{Z})(cb=\\overline{k}a) &amp;\\text{; Ex(2), clausura: }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; Def. divisibilitatis (3)}\\\\\n\n(5)&amp; \\vdash a\\mid b \\rightarrow \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; TD(4)}\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<\/ul>\n<h3><b>Theorema:<\/b> cota divisorii<\/h3>\n<p style=\"text-align: justify;\">\nSi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> et <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>.\n<\/p>\n<p><b>Demonstratio:<\/b><\/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;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash b \\neq 0\n\n&amp; \\text{; Praesumptio} \\\\\n\n(2) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (b=ka)\n\n&amp; \\text{; Def. divisibilitatis, Praesumptio} \\\\\n\n(3) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (|b|=|k||a|)\n\n&amp; \\text{; Prop. valoris absoluti, Ex(2)} \\\\\n\n(4) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (k\\neq 0 \\wedge |b|=|k||a|)\n\n&amp; \\text{; Ex(1,3)} \\\\\n\n(5) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash (\\exists k \\in \\mathbb{Z}) (1\\le |k| \\wedge |b|=|k||a|)\n\n&amp; \\text{; Ex(4), si }k\\neq 0\\Rightarrow |k|\\ge 1 \\\\\n\n(6) &amp;\\{b\\in \\mathbb{Z}\\setminus\\{0\\} , a\\mid b\\}\\vdash |a|\\le |b|\n\n&amp; \\text{; Ex(5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>Exercitationes Propositae<\/h2>\n<ol>\n<li>Ostende theorema \u201ccota divisorii\u201d non necessario verum esse si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>.<\/li>\n<li>Consideremus collectionem <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et relationem <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> super hanc collectionem. Si elementa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> ita se habeant ut <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> ad <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> referatur, tunc scribitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span>. Relatio <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> dicitur esse <strong>ordinis partialis<\/strong> super <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> si:\n<p>a)<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in A) (x\\rho x)<\/span><\/span>,<br \/>\nb) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y\\in A) ( (x\\rho y \\wedge y\\rho x) \\rightarrow x=y)<\/span><\/span><br \/>\nc) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y,z\\in A) ( (x\\rho y \\wedge y\\rho z) \\rightarrow x\\rho z)<\/span><\/span>.<\/p>\n<p>Demonstra relationem divisibilitatis esse relationem ordinis partialis super numeros integros.<\/li>\n<li>Proba per inductionem quod, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_1, a\\mid b_2, \\cdots, a\\mid b_n<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> pro quolibet coetu <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span>. Praeterea ostende quod, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span>, et <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> scribi potest ut combinatio linearis horum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span>, tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>. <\/li>\n<li>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, ostende collectionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> finitam esse.<\/li>\n<li>Considera <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span> fixum, et sit\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S=\\{d\\,:\\,d\\in\\mathbb{Z}^+ \\wedge d\\mid n\\}<\/span>\n<p>Proba:<\/p>\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\in S \\leftrightarrow n\/d\\in S<\/span><\/span><\/li>\n<li type=\"a\">Si elementa <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> ordine crescente disponuntur: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1=d_1 \\lt d_2 \\lt \\cdots \\lt d_t =n<\/span><\/span>, tunc elementa correspondentia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> ordine decrescente sunt.<\/li>\n<\/ol>\n<\/li>\n<li>Suppone <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\">ab=c<\/span><\/span>. Demonstra <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>.<\/li>\n<li>Integer <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> dicitur par si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>, et impar si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>. Demonstra summam et differentiam:\n<ol>\n<li type=\"a\">duorum parium esse numerum parem.<\/li>\n<li type=\"a\">duorum imparium esse numerum parem.<\/li>\n<li type=\"a\">pari et impari esse numerum imparem.<\/li>\n<\/ol>\n<\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> est impar a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span> distinctus, proba <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> duos pares consecutivos dividere non posse.<\/li>\n<li>Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> tales ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>. Proba <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> neque <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> neque <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> dividere posse.<\/li>\n<li>Suppone <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>. Proba:\n<ol>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall n \\in \\mathbb{Z})(a\\mid n) \\leftrightarrow a=\\pm 1<\/span><\/span><\/li>\n<li type=\"a\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall n \\in \\mathbb{Z})(n\\mid a) \\leftrightarrow a=0<\/span><\/span><\/li>\n<\/ol>\n<\/li>\n<li>Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> et supponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>. Ostende <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> implicare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>. <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Divisibilitas Divisibilitas est verum initium theoriae numerorum, quia numeros integros in systema structura praeditum convertit: iam non spectas numeros ut \u201cquantitates\u201d, sed ut partes quae inter se congruunt aut non congruunt. Una sola structura, , sinit te exprimere tam criteria simplificationis et factorizationis quam ipsum cor quorumdam processuum, ut algorithmus Euclidis, qui efficit ut maximi [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35642,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[1298,1416],"tags":[],"class_list":["post-35691","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>Quid est divisibilitas? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Quid est divisibilitas in theoria numerorum? 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