{"id":216690,"date":"2022-06-13T17:14:44","date_gmt":"2022-06-13T15:14:44","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=216690"},"modified":"2025-09-29T13:32:50","modified_gmt":"2025-09-29T11:32:50","slug":"definition-la-division-euclidienne","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/definition-la-division-euclidienne\/","title":{"rendered":"Qu\u2019est-ce que la division euclidienne ?"},"content":{"rendered":"\n

En arithm\u00e9tique, la division euclidienne<\/strong> (aussi appel\u00e9e division enti\u00e8re) est un calcul math\u00e9matique qui consiste \u00e0 diviser deux nombres entiers<\/strong> (non nuls). Ces nombres sont appel\u00e9s \u00ab\u00a0dividende\u00a0\u00bb (a) et \u00ab\u00a0diviseur\u00a0\u00bb (b). L’enjeux de l’op\u00e9ration est de trouver le \u00ab\u00a0quotient\u00a0\u00bb (q) et le \u00ab\u00a0reste\u00a0\u00bb (r). <\/p>\n\n\n\n

Soient deux entiers relatifs a et b. On suppose que b \u2208 N*. On note q et r, le quotient et le reste de la division de a par b.<\/p>\n\n\n\n\nSoient \"a\in\mathbb{Z}\" et \"b\in\mathbb{N}^*\". Alors :\n

  <\/span>   <\/span>\"\[\exists ! (q,r)\in\mathbb{Z}\times \mathbb{N} \text{ tel que } a=qb+r \text{ et }0\le r<b.\]\"<\/p>\n\"q\" est appel\u00e9 quotient et \"r\" reste de la division euclidienne de \"a\" par \"b\".\n\n\n

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\ud83d\udca1\u00c0 savoir<\/p>\n<\/div>\n

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En latin, le mot \u00ab\u00a0dividende\u00a0\u00bb (dividendus) d\u00e9signe \u00ab\u00a0celui qui doit \u00eatre divis\u00e9\u00a0\u00bb. Le mot \u00ab\u00a0quotient\u00a0\u00bb (quotiens) signifie \u00ab\u00a0combien de fois\u00a0\u00bb.<\/p>\n\n <\/div>\n <\/section>\n\n\n\n

La division euclidienne te pose probl\u00e8me ? Notre professeur particulier d’alg\u00e8bre<\/a> t’explique comment ma\u00eetriser cette op\u00e9ration fondamentale. \u2702\ufe0f<\/p>\n\n\n\n

D\u00e9monstration de la division euclidienne<\/h2>\n\n\n\n

Existence<\/h3>\n\n\n\n\n\nSoit \"\mathcal{D}=\big\{a+kb,\,k\in\mathbb{Z},\,\text{tel que }a+kb\in\mathbb{N}\big\}\". \"\mathcal{D}\" est une partie non vide de \"\mathbb{N}\" (si \"a\ge 0\", \"\mathcal{D}\" contient \"a\", sinon, \"\mathcal{D}\" contient \"a-ab\"). On en d\u00e9duit que cet ensemble contient un plus petit \u00e9l\u00e9ment que l’on note \"r\". Ainsi, pour un certain \"q\in\mathbb{Z}\", on a \"a-bq=r\", soit \"a=bq+r\".\n\n\n\n
<\/div>\n\n\n\n\n\nOn suppose que \"r\ge b\". Dans ce cas, \"r-b\ge 0\" et \"r-b=a-bq-b=a-b(q+1)\", donc \"r-b\in\mathcal{D}\" et \"r-b\" est strictement plus petit que \"r\" ce qui contredit la d\u00e9finition de \"r\". On en d\u00e9duit que \"r<b\".\n\n\n\n

Unicit\u00e9<\/h3>\n\n\n\n\nSoient \"(q,r)\" et \"(q',r')\" deux couples provenant de la division euclidienne de \"a\" par \"b\".\n\n\n\n
<\/div>\n\n\n\n\nOn a donc :

  <\/span>   <\/span>\"\[a=bq+r,\;0\le r<b\;\;\;\text{et}\;\;\;a=bq'+r',\;0\le r'<b .\]\"<\/p>\n\n\n\n

<\/div>\n\n\n\n \nAlors, en soustrayant ces deux relations, on obtient : \"b(q-q')=r'-r\".\n\n\n\n
<\/div>\n\n\n\n \nDe plus, \"-b<r'-r<b\" ce qui implique que \"-1<q-q'<1\". Or \"q-q'\" est un entier relatif, donc forc\u00e9ment \"q-q'=0\" ce qui implique imm\u00e9diatement que \"q=q'\" et \"r=r'\".\n\n\n\n

Remarque sur la d\u00e9monstration de la division euclidienne<\/h3>\n\n\n\n\nEn Python, la commande \"\mathtt{a//b}\" permet d’obtenir le quotient de la division euclidienne de \"a\" par \"b\" et la commande \"\mathtt{a\% b}\" en donne le reste.\n\n\n\n
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\"livre<\/figure>\n<\/div>\n\n\n\n
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Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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\n 2.3\/5 - (41 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

En arithm\u00e9tique, la division euclidienne (aussi appel\u00e9e division enti\u00e8re) est un calcul math\u00e9matique qui consiste \u00e0 diviser deux (…)<\/p>\n","protected":false},"author":278,"featured_media":216755,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-216690","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/216690","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/278"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=216690"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/216690\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/216755"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=216690"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=216690"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=216690"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}