{"id":35687,"date":"2025-12-26T18:56:30","date_gmt":"2025-12-26T18:56:30","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35687"},"modified":"2025-12-26T18:56:30","modified_gmt":"2025-12-26T18:56:30","slug":"quest-ce-que-la-divisibilite","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/quest-ce-que-la-divisibilite\/","title":{"rendered":"Qu\u2019est-ce que la divisibilit\u00e9 ?"},"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>Divisibilit\u00e9<\/h1>\n<p style=\"text-align:center;\"><em><br \/>\nLa <strong>divisibilit\u00e9<\/strong> constitue le v\u00e9ritable point de d\u00e9part de la th\u00e9orie des nombres, car elle transforme les entiers en un syst\u00e8me dot\u00e9 de structure : on ne consid\u00e8re plus les nombres comme de simples \u00ab quantit\u00e9s \u00bb, mais comme des \u00e9l\u00e9ments qui s\u2019imbriquent ou non les uns avec les autres. \u00c0 l\u2019aide d\u2019une seule relation, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, il est possible d\u2019exprimer aussi bien des crit\u00e8res de simplification et de factorisation que le c\u0153ur de certains proc\u00e9d\u00e9s, tels que l\u2019algorithme d\u2019Euclide, qui permet de calculer des plus grands communs diviseurs en quelques secondes, m\u00eame pour des nombres de grande taille. En outre, elle constitue la base technique d\u2019id\u00e9es qui apparaissent de mani\u00e8re r\u00e9currente en math\u00e9matiques appliqu\u00e9es et en informatique : congruences, arithm\u00e9tique modulaire, validations, codes, et (ult\u00e9rieurement) cryptographie. Ma\u00eetriser la divisibilit\u00e9 revient, en essence, \u00e0 apprendre \u00e0 d\u00e9tecter des motifs invisibles dans les entiers et \u00e0 les transformer en proc\u00e9dures qui fonctionnent toujours.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><b>Objectifs d\u2019apprentissage<\/b><br \/>\n\u00c0 l\u2019issue de ce support, l\u2019\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> la relation de divisibilit\u00e9 entre nombres entiers.<\/li>\n<li><strong>Comprendre<\/strong> la d\u00e9finition de la divisibilit\u00e9 et ses propri\u00e9t\u00e9s.<\/li>\n<li><strong>D\u00e9velopper<\/strong> des d\u00e9monstrations math\u00e9matiques de r\u00e9sultats et de th\u00e9or\u00e8mes li\u00e9s \u00e0 la divisibilit\u00e9.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>INDEX DES CONTENUS<\/u><\/b><br \/>\n<a href=\"#1\">D\u00e9finition de la divisibilit\u00e9<\/a><br \/>\n<a href=\"#2\">Propri\u00e9t\u00e9s fondamentales de la divisibilit\u00e9<\/a><br \/>\n<a href=\"#3\">Exercices propos\u00e9s<\/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>D\u00e9finition de la divisibilit\u00e9<\/h2>\n<p style=\"text-align: justify;\">\nL\u2019id\u00e9e informelle de \u00ab <em>a divise b<\/em> \u00bb devient pr\u00e9cise lorsqu\u2019on l\u2019exprime comme une relation entre entiers. Nous dirons qu\u2019un entier <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> <strong>divise<\/strong> un entier <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> peut s\u2019\u00e9crire comme un multiple exact de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>. Cette d\u00e9finition constitue la base du reste de ce support, car elle transforme des \u00e9nonc\u00e9s du type \u00ab s\u2019embo\u00eete exactement \u00bb en un crit\u00e8re v\u00e9rifiable.\n<\/p>\n<p style=\"text-align: justify;\">\n<strong>D\u00e9finition.<\/strong> Soient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in\\mathbb{Z}<\/span><\/span> avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. On dit que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> divise <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span>, et l\u2019on \u00e9crit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, si et seulement s\u2019il existe un entier <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tel que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=ka<\/span><\/span>. Dans le cas contraire, on \u00e9crit <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;\">\nDans cette d\u00e9finition, le nombre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> est appel\u00e9 <strong>quotient<\/strong> (ou facteur) associ\u00e9 \u00e0 la divisibilit\u00e9. Par exemple, affirmer <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">6\\mid 42<\/span><\/span> \u00e9quivaut \u00e0 affirmer qu\u2019il existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tel que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">42=6k<\/span><\/span> ; dans ce cas, il suffit de prendre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=7<\/span><\/span>.\n<\/p>\n<h3>Il est important de consid\u00e9rer<\/h3>\n<ul>\n<li>\n    La condition <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span> est essentielle. En effet, si l\u2019on tentait d\u2019autoriser <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=0<\/span><\/span>, la condition de divisibilit\u00e9 exigerait qu\u2019il existe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> tel que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0\\cdot k<\/span><\/span>. Or <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\cdot k=0<\/span><\/span> pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span>, de sorte que la seule possibilit\u00e9 serait <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span>. Dans ce cas, il n\u2019y aurait pas de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k<\/span><\/span> \u00ab d\u00e9termin\u00e9 \u00bb par la relation, puisque tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k\\in\\mathbb{Z}<\/span><\/span> satisfait <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=0\\cdot k<\/span><\/span>. Autrement dit, l\u2019expression informelle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=b\/a<\/span><\/span> devient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0\/0<\/span><\/span>, ce qui n\u2019est pas d\u00e9fini. Afin d\u2019\u00e9viter cette d\u00e9g\u00e9n\u00e9rescence (o\u00f9 la notion de quotient cesse d\u2019\u00eatre significative), on impose <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>. Pour cette raison, la relation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\mid b<\/span><\/span> n\u2019est pas consid\u00e9r\u00e9e comme valide.\n<\/li>\n<li>\n        En revanche, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid 0<\/span><\/span> est vrai pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span> avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span><\/span>, car il suffit de prendre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k=0<\/span><\/span> et l\u2019on a bien <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;\">\nDe cette d\u00e9finition d\u00e9coule une \u00e9quivalence que nous utiliserons de mani\u00e8re r\u00e9currente : dire que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span> revient \u00e0 dire que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> appartient \u00e0 l\u2019ensemble des multiples entiers de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span>, c\u2019est-\u00e0-dire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b\\in a\\mathbb{Z}<\/span><\/span>, o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mathbb{Z}=\\{ak:\\,k\\in\\mathbb{Z}\\}<\/span><\/span>. Cette mani\u00e8re de l\u2019\u00e9crire souligne que la divisibilit\u00e9 n\u2019est pas un \u00ab artifice \u00bb, mais une fa\u00e7on de d\u00e9crire des sous-ensembles hautement structur\u00e9s \u00e0 l\u2019int\u00e9rieur de <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>Propri\u00e9t\u00e9s fondamentales de la divisibilit\u00e9<\/h2>\n<ul>\n<li><strong>R\u00e9flexivit\u00e9 :<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid a<\/span><\/span>.<br \/>\n<u>D\u00e9monstration<\/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{; \u00c9l\u00e9ment neutre multiplicatif dans $\\mathbb{Z}$}\\\\\n\n(2)&amp;\\vdash(\\exists k \\in \\mathbb{Z})(a=ka) &amp;\\text{; Intro. existentielle (1)}\\\\\n\n(3) &amp;\\vdash a \\mid a &amp;\\text{; D\u00e9f. de la divisibilit\u00e9 (2)} \\\\\n\n&amp;\\blacksquare &amp;\n\n\\end{array}\n\n<\/span>\n<\/li>\n<li><strong>Transitivit\u00e9 :<\/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>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid c<\/span><\/span>.\n<p><u>D\u00e9monstration<\/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{; D\u00e9f. de la divisibilit\u00e9, pr\u00e9somption}\\\\\n\n(2)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash (\\exists k_2\\in\\mathbb{Z})(c=k_2b)  &amp;\\text{; D\u00e9f. de la divisibilit\u00e9, pr\u00e9somption}\\\\\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{; $\\exists$-compactification (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{Alg\u00e8bre \u00e0 l\u2019int\u00e9rieur du quantificateur}&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{Cl\u00f4ture de $\\mathbb{Z}$ pour la multiplication}&amp; \\\\\n\n(7)&amp; \\{a\\mid b ,  b\\mid c\\} \\vdash a\\mid c  &amp;\\text{; D\u00e9f. de la divisibilit\u00e9 (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>Compatibilit\u00e9 avec l\u2019addition et la soustraction :<\/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>, alors <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>D\u00e9monstration<\/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{; D\u00e9f. de la divisibilit\u00e9, pr\u00e9somption}\\\\\n\n(2)&amp;\\{a\\mid b, a\\mid c\\}\\vdash (\\exists k_2 \\in \\mathbb{Z})(c=k_2 a) &amp;\\text{; D\u00e9f. de la divisibilit\u00e9, pr\u00e9somption}\\\\\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{; $\\exists$-compactification (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{Alg\u00e8bre \u00e0 l\u2019int\u00e9rieur du quantificateur.}&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{Cl\u00f4ture de $\\mathbb{Z}$ pour l\u2019addition.}&amp; \\\\\n\n(6)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b+c) &amp;\\text{; D\u00e9f. de la divisibilit\u00e9 (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{Alg\u00e8bre \u00e0 l\u2019int\u00e9rieur du quantificateur.}&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{Cl\u00f4ture de $\\mathbb{Z}$ pour la soustraction.}&amp; \\\\\n\n(10)&amp;\\{a\\mid b, a\\mid c\\}\\vdash a\\mid (b-c) &amp;\\text{; D\u00e9f. de la divisibilit\u00e9 (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{; $\\wedge$-intro. dans le cons\u00e9quent (7,11) }\\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}<\/span>\n<\/li>\n<li><strong>Compatibilit\u00e9 avec les produits :<\/strong> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b<\/span><\/span>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid (bc)<\/span><\/span> pour tout <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\in\\mathbb{Z}<\/span><\/span>.<br \/>\n<u>D\u00e9monstration<\/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{; D\u00e9f. de la divisibilit\u00e9, pr\u00e9somption}\\\\\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), $\\forall$-intro. (c arbitraire)}\\\\\n\n&amp;\\text{Alg\u00e8bre dans }\\mathbb{Z}\\text{ \u00e0 l\u2019int\u00e9rieur du quantificateur 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{; De (2), cl\u00f4ture : }\\overline{k}=ck\\\\\n\n(4)&amp; \\{a\\mid b\\}\\vdash \\left(\\forall c \\in \\mathbb{Z}\\right) (a \\mid cb) &amp;\\text{; D\u00e9f. de la divisibilit\u00e9 (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>Th\u00e9or\u00e8me :<\/b> borne du diviseur<\/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>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a|\\le |b|<\/span><\/span>.\n<\/p>\n<p><b>D\u00e9monstration :<\/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{; Pr\u00e9somption} \\\\\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{; D\u00e9f. de la divisibilit\u00e9, pr\u00e9somption} \\\\\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. de la valeur absolue, 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), 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{; De (5)} \\\\\n\n&amp;\\blacksquare&amp;\n\n\\end{array}\n\n<\/span>\n<p><a name=\"3\"><\/a><\/br><\/p>\n<h2>Exercices propos\u00e9s<\/h2>\n<ol>\n<li>Montrez que le th\u00e9or\u00e8me \u00ab borne du diviseur \u00bb n\u2019est pas n\u00e9cessairement vrai si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b=0<\/span><\/span><\/li>\n<li>Consid\u00e9rons un ensemble <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et une relation <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> sur cet ensemble. Si les \u00e9l\u00e9ments <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span> sont tels que <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> est reli\u00e9 \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>, alors on \u00e9crit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\rho y<\/span><\/span><\/span>. La relation <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> est dite <strong>d\u2019ordre partiel<\/strong> sur <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>D\u00e9montrez que la relation de divisibilit\u00e9 est une relation d\u2019ordre partiel sur les entiers.<\/li>\n<li>D\u00e9montrez par induction que 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>, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid \\sum_{i=1}^n b_i x_i<\/span><\/span> pour tout ensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x_i\\}_{i=1}^n \\subset \\mathbb{Z}<\/span><\/span>. De plus, d\u00e9montrez que si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\mid b_i<\/span><\/span>, avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i\\in \\{1,2,3,\\cdots, n\\}<\/span><\/span> et que <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> peut \u00eatre \u00e9crit comme une combinaison lin\u00e9aire de ces <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b_i<\/span><\/span>, alors <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>, montrez que l\u2019ensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{x\\;:\\; d\\mid a\\}<\/span><\/span> est un ensemble fini.<\/li>\n<li>Consid\u00e9rez un <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{Z}^+<\/span><\/span> fix\u00e9, et soit\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>D\u00e9montrez :<\/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 les \u00e9l\u00e9ments de <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> sont dispos\u00e9s par ordre croissant : <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>, alors les \u00e9l\u00e9ments correspondants <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n\\mid d_i<\/span><\/span> avec <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i \\in \\{1,2,\\cdots, t\\}<\/span><\/span> sont en ordre d\u00e9croissant.<\/li>\n<\/ol>\n<\/li>\n<li>Supposons que <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>. D\u00e9montrez que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\min\\{a,b\\}\\le \\sqrt{c}<\/span><\/span>.<\/li>\n<li>Un entier <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> est dit pair si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\mid n<\/span><\/span>, et impair si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2\\nmid n<\/span><\/span>. D\u00e9montrez que la somme et la diff\u00e9rence de :\n<ol>\n<li type=\"a\">deux nombres pairs est un nombre pair.<\/li>\n<li type=\"a\">deux nombres impairs est un nombre pair.<\/li>\n<li type=\"a\">d\u2019un nombre pair et d\u2019un nombre impair est un nombre impair.<\/li>\n<\/ol>\n<\/li>\n<li>Si <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> est un nombre impair distinct de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\pm 1<\/span><\/span>, d\u00e9montrez que <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> ne peut pas diviser deux nombres pairs cons\u00e9cutifs.<\/li>\n<li>Soient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n\\in\\mathbb{Z}<\/span><\/span> tels que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|a-b|\\lt |n|<\/span><\/span>. D\u00e9montrez que <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> ne peut diviser ni <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> ni <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>.<\/li>\n<li>Supposons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{Z}<\/span><\/span>. D\u00e9montrez 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>Soient <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{Z}<\/span><\/span> et supposons que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0<\/span><\/span>. Montrez que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac\\mid bc<\/span><\/span> implique <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>Divisibilit\u00e9 La divisibilit\u00e9 constitue le v\u00e9ritable point de d\u00e9part de la th\u00e9orie des nombres, car elle transforme les entiers en un syst\u00e8me dot\u00e9 de structure : on ne consid\u00e8re plus les nombres comme de simples \u00ab quantit\u00e9s \u00bb, mais comme des \u00e9l\u00e9ments qui s\u2019imbriquent ou non les uns avec les autres. \u00c0 l\u2019aide d\u2019une seule [&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":5,"footnotes":""},"categories":[569,1414],"tags":[],"class_list":["post-35687","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-theorie-des-nombres"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Qu\u2019est-ce que la divisibilit\u00e9 ? - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Qu\u2019est-ce que la divisibilit\u00e9 en th\u00e9orie des nombres ? 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