{"id":3854,"date":"2020-10-21T10:32:16","date_gmt":"2020-10-21T08:32:16","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=3854"},"modified":"2020-10-21T10:32:18","modified_gmt":"2020-10-21T08:32:18","slug":"congruences-et-critere-de-divisibilite","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/","title":{"rendered":"Congruences et crit\u00e8re de divisibilit\u00e9"},"content":{"rendered":"\n<p>Congruences et crit\u00e8re de divisibilit\u00e9: pourquoi fait-on la somme des chiffres d&#8217;un nombre pour voir s&#8217;il est divisible par 3 ? Comment voir si un nombre est divisible par 11 ? Et par 7 ? La r\u00e9ponse est dans cet article&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/#Notions_de_congruences\" >Notions de congruences<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/#Congruences_et_critere_de_divisibilite_par_3\" >Congruences et crit\u00e8re de divisibilit\u00e9 par 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/#Congruences_et_critere_de_divisibilite_par_11\" >Congruences et crit\u00e8re de divisibilit\u00e9 par 11<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/#Par_7\" >Par 7<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/#Par_13\" >Par 13<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/10\/21\/congruences-et-critere-de-divisibilite\/#Par_17%E2%80%A6_euh_non_merci\" >Par 17&#8230; euh, non merci !<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Notions_de_congruences\"><\/span>Notions de congruences<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Pour la suite, vous aurez besoin  de comprendre ce qu&#8217;est une <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Congruence_sur_les_entiers\" target=\"_blank\" rel=\"noreferrer noopener\">congruence<\/a>. Voici les indispensables.<\/p>\n\n\n\n<p>Si <em>a<\/em> et <em>b<\/em> sont deux nombres entiers, avec <em>a<\/em> > <em>b<\/em>, la division euclidienne de <em>a<\/em> par <em>b<\/em> s&#8217;\u00e9crit:$$a = bq + r\\quad,\\quad 0 \\leqslant r &lt; b.$$<\/p>\n\n\n\n<p>On dira alors que <em>a<\/em> est <em>congru<\/em> \u00e0 <em>r<\/em> <em>modulo<\/em> <em>b<\/em>, et on \u00e9crira:$$a\\equiv r \\mod b.$$C&#8217;est \u00e7a une congruence.<\/p>\n\n\n\n<p>Un r\u00e9sultat important est le suivant: si <em>k<\/em> est un entier naturel, $$a\\equiv r\\mod b \\Rightarrow a^k \\equiv r^k \\mod b.$$Nous nous en servirons pour le <a href=\"#crit-trois\">crit\u00e8re de divisibilit\u00e9 par 3<\/a> et 9.<\/p>\n\n\n\n<p>Un autre r\u00e9sultat important est que : si  <em>a<\/em> &lt; <em>b<\/em> et si <em>b<\/em> = <em>a<\/em> + <em>r<\/em>, alors:$$a\\equiv -r\\mod b.$$Par exemple, $$10=11-1$$donc:$$10\\equiv-1\\mod11.$$Nous nous en servirons  pour \u00e9tablir le <a href=\"#crit-onze\">crit\u00e8re de divisibilit\u00e9 par 11<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"crit-trois\"><span class=\"ez-toc-section\" id=\"Congruences_et_critere_de_divisibilite_par_3\"><\/span>Congruences et crit\u00e8re de divisibilit\u00e9 par 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Commen\u00e7ons par un crit\u00e8re connu depuis le coll\u00e8ge: <\/p>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>Un nombre est divisible par 3 quand la somme de ses chiffres l&#8217;est aussi.<\/p><cite>Crit\u00e8re de divisibilit\u00e9 par 3<\/cite><\/blockquote><\/figure>\n\n\n\n<p>Notons:$$N = \\sum_{k=0}^n x_k\\times10^k$$c&#8217;est-\u00e0-dire:$$N = x_n\\times10^n + x_{n-1}\\times10^{n-1}+cdots+10x_1 +x_0.$$Par exemple,$$148=100 + 40 + 8 = 1\\times10^2 + 4\\times10^1 + 8\\times10^0.$$<\/p>\n\n\n\n<p>Nous savons que:$$10=3\\times3+1$$et donc que:$$10\\equiv1\\mod3$$et donc que pour tout entier naturel <em>k<\/em>:$$10^k\\equiv1\\mod3.$$<\/p>\n\n\n\n<p>Ainsi, $$N\\equiv\\sum_{k=0}^n x_k\\mod3,$$autrement dit:$$N\\equiv x_n+x_{n-1}+\\cdots+x_1+x_0\\mod3.$$<\/p>\n\n\n\n<p>On a alors:$$\\begin{align}N\\text{ divisible par 3} &amp; \\iff N \\equiv 0 \\mod 3\\\\&amp;\\iff x_n+x_{n-1}+\\cdots+x_1+x_0\\equiv0\\mod3  \\end{align}$$ce qui signifie que <em>N<\/em> est divisible par 3 si et seulement si la somme de ses chiffres l&#8217;est aussi.<\/p>\n\n\n\n<p>Au passage, on remarquera que remplacer &#8220;3&#8221; par &#8220;9&#8221; ne change rien \u00e0 la d\u00e9monstration, ce qui signifie le r\u00e9sultat suivant:<\/p>\n\n\n\n<figure class=\"wp-block-pullquote\"><blockquote><p>Un nombre est divisible par 9 si la somme de ses chiffres l&#8217;est aussi.<\/p><cite>Crit\u00e8re de divisibilit\u00e9 par 9<\/cite><\/blockquote><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"crit-onze\"><span class=\"ez-toc-section\" id=\"Congruences_et_critere_de_divisibilite_par_11\"><\/span>Congruences et crit\u00e8re de divisibilit\u00e9 par 11<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Les notations sont les m\u00eames que pr\u00e9c\u00e9demment.<\/p>\n\n\n\n<p>Nous savons que:$$10\\equiv-1\\mod11$$donc nous pouvons \u00e9crire:$$\\begin{align}N &amp; \\equiv\\sum_{k=0}^n x_k\\times(-1)^k\\mod11\\\\&amp;\\equiv\\sum_{k\\text{ pair}}x_k + \\sum_{k\\text{ impair}}(-1)x_k\\mod11\\\\&amp;\\equiv \\sum_{k\\text{ pair}}x_k &#8211; \\sum_{k\\text{ impair}}x_k\\mod11\\end{align}$$<\/p>\n\n\n\n<p>Ainsi,$$\\begin{align}N\\text{ divisible par 11} &amp; \\iff N \\equiv 0 \\mod 11\\\\&amp;\\iff  \\sum_{k\\text{ pair}}x_k &#8211; \\sum_{k\\text{ impair}}x_k\\equiv0\\mod11\\end{align}$$<\/p>\n\n\n\n<p>Cela peut para\u00eetre compliqu\u00e9 au prime abord, mais c&#8217;est tr\u00e8s simple au final. Regardons sur un exemple. Soit:$$N=5025449.$$<\/p>\n\n\n\n<p>Le crit\u00e8re de divisibilit\u00e9 que nous venons de trouver sugg\u00e8re d&#8217;ajouter tous les chiffres de rangs pairs (notons P cette somme) et tout ceux de rangs impairs (notons I cette somme).$$\\begin{align}P=9+4+2+5=20\\\\I=4+5+0=9\\end{align}$$<\/p>\n\n\n\n<p>\\(P-I=20-9=11\\equiv0\\mod11\\) donc \\(N\\equiv0\\mod11\\), et donc <em>N<\/em> est divisible par 11.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Par_7\"><\/span>Par 7<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Les notions sont les m\u00eames que pr\u00e9c\u00e9demment. Nous allons justement nous inspirer de la technique pr\u00e9c\u00e9dente en \u00e9crivant, dans un premier temps:$$10\\equiv-4\\mod7$$car \\(10\\equiv3\\mod7\\) donc \\(10\\equiv3-7\\mod7\\). Ainsi,$$\\begin{align}N &amp; \\equiv \\sum_{k=0}^n (-4)^kx_k\\mod7 \\\\ &amp; \\equiv\\sum_{k=0}^n(-1)^k 4^kx_k\\mod7.\\end{align}$$Notons maintenant que:$$\\begin{align}4^0 &amp; \\equiv 1 \\mod 7\\\\4^1 &amp; \\equiv 4 \\mod 7\\\\4^2 &amp; \\equiv2\\mod7\\\\4^3 &amp; \\equiv 1\\mod7\\end{align}$$<\/p>\n\n\n\n<p>Il y a donc un &#8220;cycle&#8221; dans les puissances de 4 modulo 7, ce qui nous permet d&#8217;\u00e9crire:$$N\\equiv x_0-4x_1+2x_2-x_3+4x_4-2x_5+x_6-4x_7+\\cdots\\mod7$$<\/p>\n\n\n\n<p>Ainsi, <em>N<\/em> est divisible par 7 si \\(x_0-4x_1+2x_2-x_3+4x_4-2x_5+x_6-4x_7+\\cdots\\equiv0\\mod7\\).<\/p>\n\n\n\n<p>Je vous l&#8217;accorde, celui-l\u00e0, il est cool&#8230; \ud83d\ude42 Regardons sur un exemple en prenant:$$N=379995.$$On calcule:$$x_0-4x_1+2x_2-x_3+4x_4-2x_5+x_6.$$Pour ma part, je pr\u00e9sente les calculs en tableau:<\/p>\n\n\n\n<figure class=\"wp-block-table aligncenter\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">signes<\/th><th class=\"has-text-align-center\" data-align=\"center\">coef.<\/th><th class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/th><th class=\"has-text-align-center\" data-align=\"center\">chiffres<\/th><th class=\"has-text-align-center\" data-align=\"center\">=<\/th><th class=\"has-text-align-center\" data-align=\"center\">r\u00e9sultats<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">-36<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">18<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">-9<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">28<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">-6<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>Quand on ajoute les r\u00e9sultats (derni\u00e8re colonne), on trouve 0 (qui est bien congru \u00e0 0 modulo 7), donc <em>N<\/em> est bien divisible par 7.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Par_13\"><\/span>Par 13<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Nous allons nous inspirer de ce qui a \u00e9t\u00e9 fait pour 7. En effet,$$10\\equiv-3\\mod13$$et:$$\\begin{align}3^0\\equiv1\\mod13\\\\3^1\\equiv3\\mod13\\\\3^2\\equiv9\\mod13\\\\3^3\\equiv1\\mod13\\end{align}$$donc:$$N\\equiv x_0-3x_1+9x_2-x_3+3x_4-9x_5+x_6-3x_7+\\cdots\\mod13$$<\/p>\n\n\n\n<p>Ainsi, $$N\\equiv0\\mod13\\iff x_0-3x_1+9x_2-x_3+3x_4-9x_5+x_6-3x_7+\\cdots\\equiv0\\mod13.$$<\/p>\n\n\n\n<p>Prenons <em>N<\/em> = 5888766 et pr\u00e9sentons les calculs dans un tableau comme pr\u00e9c\u00e9demment:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">signes<\/th><th class=\"has-text-align-center\" data-align=\"center\">coef.<\/th><th class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/th><th class=\"has-text-align-center\" data-align=\"center\">chiffres<\/th><th class=\"has-text-align-center\" data-align=\"center\">=<\/th><th class=\"has-text-align-center\" data-align=\"center\">r\u00e9sultats<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">-18<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">63<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">-8<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">24<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">-72<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">+<\/td><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">\u00d7<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">=<\/td><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>On trouve 98 &#8211; 98 (en ajoutant les positifs d&#8217;une part, les n\u00e9gatifs d&#8217;autre part), donc la somme est bien congrue \u00e0 0 modulo 13, et donc <em>N<\/em> est bien divisible par 13.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Par_17%E2%80%A6_euh_non_merci\"><\/span>Par 17&#8230; euh, non merci !<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Si l&#8217;on raisonne de m\u00eame avec 17, on a:$$N=\\sum_{k=0}^n(-1)^k7^k \\mod17$$ et$$\\begin{align}7^0&amp;\\equiv1 \\mod 17\\\\7^1&amp;\\equiv7 \\mod 17\\\\7^2&amp;\\equiv15 \\mod 17\\\\7^3&amp;\\equiv3 \\mod 17\\\\7^4&amp;\\equiv4 \\mod 17\\\\7^5&amp;\\equiv11 \\mod 17\\\\7^6&amp;\\equiv9 \\mod 17\\\\7^7&amp;\\equiv12 \\mod 17\\\\7^8&amp;\\equiv16 \\mod 17\\\\7^9&amp;\\equiv10 \\mod 17\\\\7^{10}&amp;\\equiv2 \\mod 17\\\\7^{11}&amp;\\equiv14 \\mod 17\\\\7^{12}&amp;\\equiv13 \\mod 17\\\\7^{13}&amp;\\equiv6 \\mod 17\\\\7^{14}&amp;\\equiv8 \\mod 17\\\\7^{15}&amp;\\equiv5 \\mod 17\\\\7^{16}&amp;\\equiv1 \\mod 17\\end{align}$$<\/p>\n\n\n\n<p>Le fait que le cycle soit sup\u00e9rieur \u00e0 3 est probl\u00e9matique car il faudrait retenir en l&#8217;occurrence 16 coefficients (contre 3 pour la divisibilit\u00e9 par 7 ou 13).<\/p>\n\n\n\n<p>Comprenez donc que je n&#8217;insiste pas&#8230; <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Congruences et crit\u00e8re de divisibilit\u00e9: pourquoi fait-on la somme des chiffres d&#8217;un nombre pour voir s&#8217;il est divisible par 3 ? Comment voir si un nombre est divisible par 11 ? Et par 7 ? La r\u00e9ponse est dans cet article&#8230;<\/p>\n","protected":false},"author":1,"featured_media":3870,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[247,171],"class_list":["post-3854","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-congruence","tag-divisibilite"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Congruences et crit\u00e8re de divisibilit\u00e9 - Mathweb.fr - Par 3, 11, 13, 7 et... 17?<\/title>\n<meta name=\"description\" content=\"Congruences et crit\u00e8re de divisibilit\u00e9: pourquoi fait-on la somme des chiffres d&#039;un nombre pour voir s&#039;il est divisible par 3? 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