Quiz — 1586536146_17-Résolution d'un systéme de congruences.pdf

Question 1 sur 10 20:00

[{"id":114844,"question":"Quelle est la condition nécessaire pour appliquer le théorème des restes chinois à un système de congruences ?","option_a":"Les modules doivent être égaux","option_b":"Les modules doivent être premiers entre eux deux à deux","option_c":"Les modules doivent être impairs","option_d":"Les modules doivent être pairs","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des restes chinois s'applique uniquement si les modules sont deux à deux premiers entre eux. Cette condition est essentielle pour garantir l'existence d'une solution unique modulo le produit des modules.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Les modules doivent être égaux\", \"b\": \"Les modules doivent êtr","_debug_options_count":4},{"id":114845,"question":"Soit x ≡ 2 [3] et x ≡ 3 [5]. Quelle est la solution modulo 15 ?","option_a":"x ≡ 8 [15]","option_b":"x ≡ 13 [15]","option_c":"x ≡ 3 [15]","option_d":"x ≡ 5 [15]","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant le théorème des restes chinois, on trouve x = 8 comme solution unique modulo 15. Vérifiez : 8 ≡ 2 [3] et 8 ≡ 3 [5].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x ≡ 8 [15]\", \"b\": \"x ≡ 13 [15]\", \"c\": \"x ≡ 3 [15]\", \"d\": \"x","_debug_options_count":4},{"id":114846,"question":"Vrai ou Faux : Si x ≡ a [n] et x ≡ b [m], alors x ≡ a + b [n + m].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette affirmation est fausse. La somme des modules n'a aucun lien avec la solution des congruences. Par exemple, x ≡ 1 [2] et x ≡ 1 [3] ont pour solution x ≡ 1 [6], mais 1 + 1 ≠ 1 [5].","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":114847,"question":"Quelle est la solution générale de x ≡ 5 [7] ?","option_a":"x = 5 + 7k, où k ∈ ℤ","option_b":"x = 7k, où k ∈ ℤ","option_c":"x = 5k, où k ∈ ℤ","option_d":"x = 5 + k, où k ∈ ℤ","option_e":"","option_f":"","bonne_reponse":"a","explication":"La solution générale d'une congruence x ≡ a [n] est x = a + nk, où k est un entier relatif. Ici, x = 5 + 7k.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x = 5 + 7k, où k ∈ ℤ\", \"b\": \"x = 7k, où k ∈ ℤ\", \"c\": \"x","_debug_options_count":4},{"id":114848,"question":"Vrai ou Faux : Le théorème des restes chinois garantit toujours l'existence d'une solution.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une solution existe uniquement si les congruences sont compatibles, c'est-à-dire si a ≡ b [pgcd(n, m)] pour chaque paire de congruences. Sinon, le système n'a pas de solution.","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":114849,"question":"Soit le système x ≡ 1 [4] et x ≡ 3 [6]. Ce système a-t-il une solution ?","option_a":"Oui","option_b":"Non","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car 1 ≢ 3 [pgcd(4, 6)] = 2. Les congruences ne sont pas compatibles, donc le système n'a pas de solution.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Oui\", \"b\": \"Non\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":114850,"question":"Quelle est la solution de x ≡ 2 [5] et x ≡ 3 [7] ?","option_a":"x ≡ 17 [35]","option_b":"x ≡ 12 [35]","option_c":"x ≡ 23 [35]","option_d":"x ≡ 8 [35]","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant le théorème des restes chinois, on trouve x = 17 comme solution unique modulo 35. Vérifiez : 17 ≡ 2 [5] et 17 ≡ 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\": \"x ≡ 17 [35]\", \"b\": \"x ≡ 12 [35]\", \"c\": \"x ≡ 23 [35]\", \"d\": ","_debug_options_count":4},{"id":114851,"question":"Vrai ou Faux : Si x ≡ a [n] et y ≡ b [m], alors x + y ≡ a + b [pgcd(n, m)].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La somme des congruences ne se fait pas modulo le pgcd des modules, mais modulo le produit des modules si ceux-ci sont 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},{"id":114852,"question":"Quelle méthode utiliser pour résoudre x ≡ 4 [9] et x ≡ 7 [12] ?","option_a":"Méthode de substitution","option_b":"Théorème des restes chinois","option_c":"Résolution graphique","option_d":"Factorisation","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème des restes chinois est la méthode la plus adaptée pour résoudre ce système, car les modules 9 et 12 ne sont pas premiers entre eux (pgcd = 3). Il faut d'abord vérifier la compatibilité des congruences.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Méthode de substitution\", \"b\": \"Théorème des restes chinois\", ","_debug_options_count":4},{"id":114853,"question":"Soit x ≡ 0 [2] et x ≡ 1 [4]. Ce système a-t-il une solution ?","option_a":"Oui","option_b":"Non","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, car 0 ≢ 1 [pgcd(2, 4)] = 2. Les congruences ne sont pas compatibles, donc le système n'a pas de solution.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Oui\", \"b\": \"Non\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]