Quiz — Lycée Pilote Bizerte - Arithmétiques 3.pdf

Question 1 sur 10 20:00

[{"id":41190,"question":"Quel est le PGCD de 48 et 18 ?","option_a":"6","option_b":"12","option_c":"18","option_d":"24","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le PGCD de 48 et 18 se calcule par l'algorithme d'Euclide : 48 = 2×18 + 12, puis 18 = 1×12 + 6, et enfin 12 = 2×6 + 0. Le dernier reste non nul est 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"6\", \"b\": \"12\", \"c\": \"18\", \"d\": \"24\"}}","_debug_options_count":4},{"id":41191,"question":"Si a ≡ b [n], alors n divise (a - b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, a ≡ b [n] signifie que n divise (a - b). Cette propriété est fondamentale en arithmétique modulaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41192,"question":"Quel est le PPCM de 12 et 15 ?","option_a":"30","option_b":"60","option_c":"90","option_d":"120","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le PPCM se calcule à partir de la décomposition en facteurs premiers : 12 = 2²×3 et 15 = 3×5. Le PPCM est donc 2²×3×5 = 60.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"30\", \"b\": \"60\", \"c\": \"90\", \"d\": \"120\"}}","_debug_options_count":4},{"id":41193,"question":"Le théorème de Bézout s'applique uniquement aux nombres premiers.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème de Bézout s'applique à tous les entiers non nuls. Il stipule qu'il existe des entiers u et v tels que au + bv = PGCD(a, b), quel que soit le type de nombres.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41194,"question":"Si a et b sont premiers entre eux, alors PGCD(a, b) = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux nombres sont premiers entre eux si leur PGCD est égal à 1. C'est une définition fondamentale en arithmétique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41195,"question":"Quel est le reste de la division euclidienne de 1234 par 5 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"4","option_e":"","option_f":"","bonne_reponse":"d","explication":"Pour trouver le reste de 1234 ÷ 5, on calcule 1234 mod 5. 1230 est divisible par 5, donc 1234 = 1230 + 4. Le reste est 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"4\"}}","_debug_options_count":4},{"id":41196,"question":"Le théorème de Gauss stipule que si a divise bc et si a est premier avec b, alors a divise c.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Gauss est un résultat clé en arithmétique : si a divise bc et si a est premier avec b, alors a divise c. C'est utile pour simplifier des équations.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41197,"question":"Quelle est la décomposition en facteurs premiers de 84 ?","option_a":"2²×3×7","option_b":"2×3²×7","option_c":"2³×3×7","option_d":"2×3×7","option_e":"","option_f":"","bonne_reponse":"a","explication":"84 = 2×42 = 2×2×21 = 2×2×3×7. La décomposition en facteurs premiers est donc 2²×3×7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2²×3×7\", \"b\": \"2×3²×7\", \"c\": \"2³×3×7\", \"d\": \"2×3×7\"}}","_debug_options_count":4},{"id":41198,"question":"Si a ≡ 0 [n], alors n divise a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La notation a ≡ 0 [n] signifie que n divise a. C'est une propriété directe de la congruence.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41199,"question":"Quel est le plus petit entier strictement positif congru à 7 modulo 3 ?","option_a":"1","option_b":"2","option_c":"4","option_d":"7","option_e":"","option_f":"","bonne_reponse":"a","explication":"7 mod 3 = 1, car 7 = 2×3 + 1. Le plus petit entier strictement positif congru à 7 modulo 3 est donc 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"4\", \"d\": \"7\"}}","_debug_options_count":4}]