Quiz — énoncé_Identité de Bézout 1 top 50-1.pdf

Question 1 sur 10 20:00

[{"id":90021,"question":"Soient a et b deux entiers relatifs non nuls. L'identité de Bézout affirme qu'il existe des entiers u et v tels que :","option_a":"au + bv = a + b","option_b":"au + bv = PGCD(a, b)","option_c":"au + bv = a × b","option_d":"au + bv = 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'identité de Bézout stipule que pour deux entiers a et b, il existe des entiers u et v tels que au + bv = PGCD(a, b).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"au + bv = a + b\", \"b\": \"au + bv = PGCD(a, b)\", \"c\": \"au + bv = a ","_debug_options_count":4},{"id":90022,"question":"Si deux entiers a et b sont premiers entre eux, alors leur PGCD est égal à :","option_a":"0","option_b":"1","option_c":"a","option_d":"b","option_e":"","option_f":"","bonne_reponse":"b","explication":"Deux entiers sont premiers entre eux si et seulement 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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"a\", \"d\": \"b\"}}","_debug_options_count":4},{"id":90023,"question":"L'identité de Bézout permet de résoudre des équations de la forme :","option_a":"ax + by = 0","option_b":"ax + by = c (c entier)","option_c":"ax² + by² = c","option_d":"a^x + b^y = c","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'identité de Bézout est utilisée pour résoudre des équations diophantiennes linéaires de la forme ax + by = c, où c est un entier.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"ax + by = 0\", \"b\": \"ax + by = c (c entier)\", \"c\": \"ax² + by² = ","_debug_options_count":4},{"id":90024,"question":"Soient a = 12 et b = 18. Leur PGCD est 6. Parmi les propositions suivantes, laquelle est une solution de l'équation 12u + 18v = 6 ?","option_a":"u = 1, v = 1","option_b":"u = 2, v = -1","option_c":"u = 0, v = 1","option_d":"u = -1, v = 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En substituant u = 2 et v = -1 dans l'équation, on obtient 12×2 + 18×(-1) = 24 - 18 = 6, ce qui correspond au PGCD.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"u = 1, v = 1\", \"b\": \"u = 2, v = -1\", \"c\": \"u = 0, v = 1\", \"d\": \"u","_debug_options_count":4},{"id":90025,"question":"Deux nombres a et b sont premiers entre eux si et seulement si l'identité de Bézout s'écrit au + bv = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si a et b sont premiers entre eux, alors leur PGCD est 1, donc l'identité de Bézout s'écrit bien au + bv = 1. La proposition est donc vraie.","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":90026,"question":"Quel est l'intérêt principal de l'identité de Bézout dans la résolution de problèmes ?","option_a":"Trouver des solutions entières à des équations","option_b":"Calculer directement le PGCD de deux nombres","option_c":"Démontrer que deux nombres sont divisibles","option_d":"Simplifier des fractions algébriques","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'identité de Bézout est principalement utilisée pour trouver des solutions entières à des équations diophantiennes linéaires.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Trouver des solutions entières à des équations\", \"b\": \"Calcule","_debug_options_count":4},{"id":90027,"question":"Si a = 5 et b = 7, alors il existe des entiers u et v tels que 5u + 7v = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Puisque 5 et 7 sont premiers entre eux, leur PGCD est 1, et l'identité de Bézout garantit l'existence de tels entiers u et v.","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":90028,"question":"Parmi les couples suivants, lequel représente une solution de l'équation 8u + 12v = 4 ?","option_a":"u = 1, v = 1","option_b":"u = 2, v = -1","option_c":"u = 0, v = 1","option_d":"u = -1, v = 2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En substituant u = 2 et v = -1 dans l'équation, on obtient 8×2 + 12×(-1) = 16 - 12 = 4, ce qui est correct.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"u = 1, v = 1\", \"b\": \"u = 2, v = -1\", \"c\": \"u = 0, v = 1\", \"d\": \"u","_debug_options_count":4},{"id":90029,"question":"L'identité de Bézout est valable uniquement pour les entiers naturels.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'identité de Bézout s'applique aux entiers relatifs (positifs ou négatifs), pas uniquement aux entiers naturels.","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":90030,"question":"Soient a = 9 et b = 15. Leur PGCD est 3. Parmi les solutions suivantes, laquelle vérifie l'équation 9u + 15v = 3 ?","option_a":"u = 2, v = -1","option_b":"u = 1, v = 0","option_c":"u = 0, v = 1","option_d":"u = -1, v = 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En substituant u = 2 et v = -1 dans l'équation, on obtient 9×2 + 15×(-1) = 18 - 15 = 3, ce qui est correct.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"u = 2, v = -1\", \"b\": \"u = 1, v = 0\", \"c\": \"u = 0, v = 1\", \"d\": \"u","_debug_options_count":4}]