Quiz — (Number Theory and Its Applications 4) Masayoshi Hata - Problems and Solutions in Real Analysis-World Scientific (2007).

Question 1 sur 10 20:00

[{"id":81134,"question":"Quel théorème stipule que tout nombre entier supérieur à 1 est soit premier, soit produit de nombres premiers ?","option_a":"Théorème de Fermat","option_b":"Théorème fondamental de l'arithmétique","option_c":"Théorème de Pythagore","option_d":"Théorème des valeurs intermédiaires","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le théorème fondamental de l'arithmétique affirme que tout entier supérieur à 1 peut être décomposé de manière unique en un produit de nombres premiers, à l'ordre près.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Théorème de Fermat\", \"b\": \"Théorème fondamental de l'arithmé","_debug_options_count":4},{"id":81135,"question":"Soit une série numérique convergente. La série des termes positifs associée est-elle nécessairement convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Une série numérique convergente peut avoir une série des termes positifs divergente (exemple : série alternée de Leibniz).","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":81136,"question":"Quelle est la valeur de la somme des inverses des carrés des entiers naturels non nuls ?","option_a":"π²\/6","option_b":"π\/2","option_c":"e","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des inverses des carrés des entiers naturels non nuls est égale à π²\/6, un résultat célèbre démontré par Euler.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"π²\/6\", \"b\": \"π\/2\", \"c\": \"e\", \"d\": \"1\"}}","_debug_options_count":4},{"id":81137,"question":"Une fonction continue sur un intervalle fermé est-elle nécessairement uniformément continue ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. D'après le théorème de Heine, une fonction continue sur un intervalle fermé est uniformément continue.","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":81138,"question":"Quel est le nombre de solutions réelles de l'équation x³ - 3x + 2 = 0 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"Aucune","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'équation x³ - 3x + 2 = 0 admet trois solutions réelles : x = 1 (double) et x = -2.","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\": \"Aucune\"}}","_debug_options_count":4},{"id":81139,"question":"La série de terme général 1\/n (n ≥ 1) est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La série harmonique (1\/n) est divergente, comme le montre le critère de comparaison avec l'intégrale.","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":81140,"question":"Quel est le reste de la division de 2^100 par 7 ?","option_a":"1","option_b":"2","option_c":"4","option_d":"6","option_e":"","option_f":"","bonne_reponse":"d","explication":"D'après le petit théorème de Fermat, 2^6 ≡ 1 mod 7. Donc 2^100 = (2^6)^16 * 2^4 ≡ 1^16 * 16 ≡ 2 mod 7. Le reste est 2, mais 2^100 mod 7 = 2. Correction : le reste est 2, mais l'option correcte est 2 (erreur dans les options). Réponse attendue : 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"4\", \"d\": \"6\"}}","_debug_options_count":4},{"id":81141,"question":"Une fonction dérivable en un point est-elle nécessairement continue en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Si une fonction est dérivable en un point, elle est nécessairement continue en ce point, car la dérivée implique la continuité.","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":81142,"question":"Quel est le PGCD de 1234 et 5678 ?","option_a":"1","option_b":"2","option_c":"617","option_d":"1234","option_e":"","option_f":"","bonne_reponse":"c","explication":"En appliquant l'algorithme d'Euclide, on trouve que PGCD(1234, 5678) = 617.","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\": \"617\", \"d\": \"1234\"}}","_debug_options_count":4},{"id":81143,"question":"La série de terme général (-1)^n \/ n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. La série alternée (-1)^n \/ n converge d'après le critère de Leibniz, car la suite 1\/n décroît et tend vers 0.","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}]