{"id":35661,"date":"2025-12-26T18:56:31","date_gmt":"2025-12-26T18:56:31","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35661"},"modified":"2025-12-26T18:56:31","modified_gmt":"2025-12-26T18:56:31","slug":"o-que-e-a-divisibilidade","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/pt\/o-que-e-a-divisibilidade\/","title":{"rendered":"O que \u00e9 a Divisibilidade?"},"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>Divisibilidade<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\nA <strong>divisibilidade<\/strong> \u00e9 o verdadeiro ponto de partida da teoria dos n\u00fameros porque transforma os inteiros em um sistema com estrutura: voc\u00ea deixa de olhar os n\u00fameros como \u201cquantidades\u201d e passa a v\u00ea-los como pe\u00e7as que se encaixam ou n\u00e3o entre si. Com uma \u00fanica estrutura, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, \u00e9 poss\u00edvel expressar desde crit\u00e9rios de simplifica\u00e7\u00e3o e fatora\u00e7\u00e3o at\u00e9 o n\u00facleo de alguns procedimentos, como o algoritmo de Euclides, que permite calcular m\u00e1ximos divisores comuns em segundos, mesmo com n\u00fameros grandes. Al\u00e9m disso, ela constitui a base t\u00e9cnica de ideias que aparecem repetidamente na matem\u00e1tica aplicada e na computa\u00e7\u00e3o: congru\u00eancias, aritm\u00e9tica modular, valida\u00e7\u00f5es, c\u00f3digos e (mais adiante) criptografia. Dominar a divisibilidade \u00e9, em ess\u00eancia, aprender a detectar padr\u00f5es invis\u00edveis nos inteiros e a convert\u00ea-los em procedimentos que funcionam sempre.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>Objetivos de Aprendizagem<\/b><br \/>\nAo final deste apunte, o estudante ser\u00e1 capaz de:\n<\/p>\n<ol>\n<li><strong>Compreender<\/strong> a rela\u00e7\u00e3o de divisibilidade entre n\u00fameros inteiros.<\/li>\n<li><strong>Compreender<\/strong> a defini\u00e7\u00e3o de divisibilidade e suas propriedades.<\/li>\n<li><strong>Desenvolver<\/strong> demonstra\u00e7\u00f5es matem\u00e1ticas de resultados e teoremas relacionados \u00e0 divisibilidade.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>\u00cdNDICE DE CONTE\u00daDOS<\/u><\/b><br \/>\n<a href=\"#1\">Defini\u00e7\u00e3o de divisibilidade<\/a><br \/>\n<a href=\"#2\">Propriedades fundamentais da divisibilidade<\/a><br \/>\n<a href=\"#3\">Exerc\u00edcios Propostos<\/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>Defini\u00e7\u00e3o de divisibilidade<\/h2>\n<p style=\"text-align: justify;\">\nA ideia informal de \u201c<em>a divide b<\/em>\u201d torna-se precisa quando a expressamos como uma rela\u00e7\u00e3o entre inteiros. Diremos que um inteiro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>divide<\/strong> um inteiro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> pode ser escrito como um m\u00faltiplo exato de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. Essa defini\u00e7\u00e3o \u00e9 a base do restante do apunte, pois transforma frases do tipo \u201cencaixa exatamente\u201d em um crit\u00e9rio verific\u00e1vel.\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>Defini\u00e7\u00e3o.<\/strong> Sejam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. Dizemos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> divide <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, e escrevemos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, se e somente se existe um inteiro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>. Caso contr\u00e1rio, escrevemos <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;\">\nNessa defini\u00e7\u00e3o, o n\u00famero <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u00e9 chamado de <strong>quociente<\/strong> (ou fator) associado \u00e0 divisibilidade. Por exemplo, afirmar <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> equivale a afirmar que existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span>; nesse caso, basta tomar <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>.\n<\/p>\n<h3>\u00c9 importante considerar<\/h3>\n<ul>\n<li>\n    A condi\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> \u00e9 essencial, pois, se tent\u00e1ssemos permitir <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>, a condi\u00e7\u00e3o de divisibilidade exigiria que existisse <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>. Contudo, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>, de modo que a \u00fanica possibilidade seria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>. Nesse caso, n\u00e3o haveria um <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u201cdeterminado\u201d pela rela\u00e7\u00e3o, j\u00e1 que qualquer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> satisfaz <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span>. Em outras palavras, a express\u00e3o informal <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> torna-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>, o que n\u00e3o est\u00e1 definido. Para evitar essa degenera\u00e7\u00e3o (em que a no\u00e7\u00e3o de quociente deixa de ser significativa), exige-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. Por esse motivo, a rela\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> n\u00e3o \u00e9 considerada v\u00e1lida.\n<\/li>\n<li>\n        Em contrapartida, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> \u00e9 verdadeiro para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, pois basta tomar <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span> e tem-se <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;\">\nA partir dessa defini\u00e7\u00e3o, decorre uma equival\u00eancia que utilizaremos de forma recorrente: dizer que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> \u00e9 o mesmo que dizer que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> pertence ao conjunto dos m\u00faltiplos inteiros de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, isto \u00e9, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>, onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>. Essa forma de escrita enfatiza que a divisibilidade n\u00e3o \u00e9 um \u201ctruque\u201d, mas uma maneira de descrever subconjuntos altamente estruturados dentro de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}<\/span><\/span>.\n<\/p>\n<p><a name=\"2\"><\/a><\/br><\/p>\n<h2>Propriedades fundamentais da divisibilidade<\/h2>\n<ul>\n<li><strong>Reflexividade:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>Demonstra\u00e7\u00e3o<\/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{; Neutro multiplicativo em $\\mathbb{Z}$}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; Int. existencial (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; Def. de divisibilidade (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>Transitividade:<\/strong> se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\mid c<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>.\n<p><u>Demonstra\u00e7\u00e3o<\/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. de divisibilidade, presun\u00e7\u00e3o}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; Def. de divisibilidade, presun\u00e7\u00e3o}\\\\\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{; $\\exists$-compacta\u00e7\u00e3o (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{; De (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{; De (4)}\\\\\n\n&amp;\\text{\u00c1lgebra dentro do quantificador}&amp; \\\\\n\n(6)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k\\in\\mathbb{Z})( c=ka)  &amp;\\text{; De (5)}\\\\\n\n&amp;\\text{Fechamento de $\\mathbb{Z}$ para a multiplica\u00e7\u00e3o}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; Def. de divisibilidade (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>Compatibilidade com soma e subtra\u00e7\u00e3o:<\/strong> se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b+c)<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (b-c)<\/span><\/span>.<br \/>\n<u>Demonstra\u00e7\u00e3o<\/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. de divisibilidade, presun\u00e7\u00e3o}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; Def. de divisibilidade, presun\u00e7\u00e3o}\\\\\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{; $\\exists$-compacta\u00e7\u00e3o (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{; De (3)}\\\\\n\n&amp;\\text{\u00c1lgebra dentro do quantificador.}&amp; \\\\\n\n(5)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k \\in \\mathbb{Z})(b+c= ka) &amp;\\text{; De (4)}\\\\\n\n&amp;\\text{Fechamento de $\\mathbb{Z}$ para a soma.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; Def. de divisibilidade (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{; De (3)}\\\\\n\n&amp;\\text{\u00c1lgebra dentro do quantificador.}&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{; De (8)}\\\\\n\n&amp;\\text{Fechamento de $\\mathbb{Z}$ para a subtra\u00e7\u00e3o.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; Def. de divisibilidade (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{; $\\wedge$-int. no consequente (7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>Compatibilidade com produtos:<\/strong> se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> para todo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span>.<br \/>\n<u>Demonstra\u00e7\u00e3o<\/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. de divisibilidade, presun\u00e7\u00e3o}\\\\\n\n(2)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (\\exists k\\in\\mathbb{Z})(cb=cka) &amp;\\text{; De (1), $\\forall$-int. (c arbitr\u00e1rio)}\\\\\n\n&amp;\\text{\u00c1lgebra em }\\mathbb{Z}\\text{ dentro do quantificador existencial.}&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{; De (2), fechamento: }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; Def. de divisibilidade (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>Teorema:<\/b> cota do divisor<\/h3>\n<p style=\"text-align: justify;\">\nSe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\neq 0<\/span><\/span> e <span dir=\"ltr\"> <span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>.\n<\/p>\n<p><b>Demonstra\u00e7\u00e3o:<\/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{; Presun\u00e7\u00e3o} \\\\\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. de divisibilidade, presun\u00e7\u00e3o} \\\\\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. do valor absoluto, De (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{; De (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{; De (4), se }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{; De (5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>Exerc\u00edcios Propostos<\/h2>\n<ol>\n<li>Mostre que o teorema \u201ccota do divisor\u201d n\u00e3o \u00e9 necessariamente verdadeiro se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span><\/li>\n<li>Consideremos um conjunto <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e uma rela\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> sobre esse conjunto. Se os elementos <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> forem tais que <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> est\u00e1 relacionado com <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> por meio de <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>, ent\u00e3o escreve-se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span><\/span>. Diz-se que a rela\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> \u00e9 <strong>de ordem parcial<\/strong> sobre <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> se:\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>Demonstre que a rela\u00e7\u00e3o de divisibilidade \u00e9 uma rela\u00e7\u00e3o de ordem parcial sobre os n\u00fameros inteiros.<\/li>\n<li>Prove por indu\u00e7\u00e3o que, se <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>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> para qualquer conjunto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span>. Al\u00e9m disso, prove que, se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>, com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> pode ser escrito como uma combina\u00e7\u00e3o linear desses <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span>, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>. <\/li>\n<li>Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, mostre que o conjunto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> \u00e9 um conjunto finito.<\/li>\n<li>Considere um <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span> fixo, e seja\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>Prove:<\/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\">Se os elementos de <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> forem colocados em ordem crescente: <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>, ent\u00e3o os elementos correspondentes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> com <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> est\u00e3o em ordem decrescente.<\/li>\n<\/ol>\n<\/li>\n<li>Suponha que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}^+<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=c<\/span><\/span>. Demonstre que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>.<\/li>\n<li>Um inteiro <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> diz-se par se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>, e \u00edmpar se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>. Demonstre que a soma e a diferen\u00e7a de:\n<ol>\n<li type=\"a\">dois pares \u00e9 um n\u00famero par.<\/li>\n<li type=\"a\">dois \u00edmpares \u00e9 um n\u00famero par.<\/li>\n<li type=\"a\">de um par e um \u00edmpar \u00e9 um n\u00famero \u00edmpar.<\/li>\n<\/ol>\n<\/li>\n<li>Se <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u00e9 um \u00edmpar distinto de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span>, prove que <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> n\u00e3o pode dividir dois pares consecutivos.<\/li>\n<li>Sejam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> tais que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>. Prove que <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> n\u00e3o pode dividir nem <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> nem <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>.<\/li>\n<li>Suponha que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>. Prove que:\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>Sejam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> e suponhamos que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>. Mostre que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> implica que <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>Divisibilidade A divisibilidade \u00e9 o verdadeiro ponto de partida da teoria dos n\u00fameros porque transforma os inteiros em um sistema com estrutura: voc\u00ea deixa de olhar os n\u00fameros como \u201cquantidades\u201d e passa a v\u00ea-los como pe\u00e7as que se encaixam ou n\u00e3o entre si. Com uma \u00fanica estrutura, , \u00e9 poss\u00edvel expressar desde crit\u00e9rios de simplifica\u00e7\u00e3o [&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":4,"footnotes":""},"categories":[571,1404],"tags":[],"class_list":["post-35661","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica-pt","category-teoria-dos-numeros"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>O que \u00e9 a Divisibilidade? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"O que \u00e9 a divisibilidade na teoria dos n\u00fameros? 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