Quiz — Lycée Pilote Sfax - Arithmétique 3.pdf

Question 1 sur 10 20:00

[{"id":37880,"question":"Quel est le PGCD de 48 et 18 ?","option_a":"2","option_b":"6","option_c":"12","option_d":"18","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le PGCD de 48 et 18 se calcule avec 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\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"6\", \"c\": \"12\", \"d\": \"18\"}}","_debug_options_count":4},{"id":37881,"question":"Deux nombres entiers a et b sont premiers entre eux si et seulement si leur PGCD vaut 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, deux nombres sont premiers entre eux si leur PGCD est égal à 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":37882,"question":"Quel est le PPCM de 12 et 18 ?","option_a":"36","option_b":"24","option_c":"18","option_d":"12","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le PPCM (Plus Petit Commun Multiple) de 12 et 18 se calcule en utilisant la formule PPCM(a,b) = (a×b)\/PGCD(a,b). Ici, PGCD(12,18)=6, donc PPCM(12,18)=(12×18)\/6=36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"36\", \"b\": \"24\", \"c\": \"18\", \"d\": \"12\"}}","_debug_options_count":4},{"id":37883,"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 de la congruence, a ≡ b [n] signifie que n divise (a - b).","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":37884,"question":"Quel est le résultat de 15 mod 4 ?","option_a":"0","option_b":"1","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le reste de la division de 15 par 4 est 3, donc 15 mod 4 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":37885,"question":"Le théorème de Gauss stipule que si un nombre premier p divise un produit ab, alors p divise a ou p divise b.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Gauss est une propriété fondamentale en arithmétique : si un nombre premier p divise ab, alors p divise a ou p divise b.","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":37886,"question":"Quel est le PGCD de 210 et 45 ?","option_a":"5","option_b":"15","option_c":"30","option_d":"45","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant l'algorithme d'Euclide : 210 = 4×45 + 30, puis 45 = 1×30 + 15, et enfin 30 = 2×15 + 0. Le dernier reste non nul est 15.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"5\", \"b\": \"15\", \"c\": \"30\", \"d\": \"45\"}}","_debug_options_count":4},{"id":37887,"question":"Si a et b sont deux entiers non nuls, alors PGCD(a,b) × PPCM(a,b) = a × b.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette égalité est une propriété fondamentale liant PGCD et PPCM : PGCD(a,b) × PPCM(a,b) = a × b.","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":37888,"question":"Quel est le résultat de 7 × 13 mod 5 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"D'abord, 7 × 13 = 91. Ensuite, 91 mod 5 = 1 (car 91 = 18×5 + 1). Cependant, 1 mod 5 = 1, donc la réponse correcte est 1. Correction : la bonne réponse est 1, mais l'explication précédente était erronée. 91 divisé par 5 donne 18 avec un reste de 1, donc 91 mod 5 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":37889,"question":"Le théorème de Bézout s'applique uniquement si a et b sont premiers entre eux.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème de Bézout stipule que pour tous entiers a et b, il existe des entiers u et v tels que au + bv = PGCD(a,b). Il ne nécessite pas que a et b soient premiers entre eux.","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}]