Quiz — cours_entiers.pdf

Question 1 sur 10 20:00

[{"id":18739,"question":"Quel est le PGCD de 120 et 45 ?","option_a":"15","option_b":"30","option_c":"5","option_d":"10","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le PGCD de 120 et 45 se calcule par l'algorithme d'Euclide : 120 = 2×45 + 30, puis 45 = 1×30 + 15, et enfin 30 = 2×15 + 0. Le PGCD est donc 15.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"15\", \"b\": \"30\", \"c\": \"5\", \"d\": \"10\"}}","_debug_options_count":4},{"id":18740,"question":"Si a et b sont premiers entre eux, alors le théorème de Bezout affirme qu'il existe des entiers u et v tels que au + bv = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition même du théorème de Bezout : si a et b sont premiers entre eux, il existe des entiers u et v vérifiant au + bv = 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":18741,"question":"Quel est le reste de la division de 1234 par 7 ?","option_a":"2","option_b":"3","option_c":"4","option_d":"5","option_e":"","option_f":"","bonne_reponse":"c","explication":"1234 ÷ 7 = 176 avec un reste de 2, car 7 × 176 = 1232 et 1234 - 1232 = 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\": \"2\", \"b\": \"3\", \"c\": \"4\", \"d\": \"5\"}}","_debug_options_count":4},{"id":18742,"question":"Si a ≡ b [n] et c ≡ d [n], alors a + c ≡ b + d [n].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette propriété est vraie : si a et c sont congrus à b et d modulo n, alors leur somme est aussi congruente modulo n.","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":18743,"question":"Quel est le plus petit entier naturel x tel que 5x ≡ 3 [7] ?","option_a":"2","option_b":"3","option_c":"4","option_d":"5","option_e":"","option_f":"","bonne_reponse":"b","explication":"On cherche x tel que 5x ≡ 3 [7]. En testant les valeurs : 5×2=10≡3 [7], donc x=2 est 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\": \"2\", \"b\": \"3\", \"c\": \"4\", \"d\": \"5\"}}","_debug_options_count":4},{"id":18744,"question":"Le théorème de Gauss stipule que si un nombre premier divise un produit ab, alors il divise a ou b.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est l'énoncé exact du théorème de Gauss : si p est premier et 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":18745,"question":"Quel est le nombre de diviseurs positifs de 60 ?","option_a":"8","option_b":"10","option_c":"12","option_d":"16","option_e":"","option_f":"","bonne_reponse":"c","explication":"60 = 2² × 3 × 5. Le nombre de diviseurs est (2+1)(1+1)(1+1) = 3×2×2 = 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"8\", \"b\": \"10\", \"c\": \"12\", \"d\": \"16\"}}","_debug_options_count":4},{"id":18746,"question":"Si a ≡ b [n] et k est un entier, alors ka ≡ kb [n].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette propriété est vraie : multiplier deux nombres congrus modulo n par un même entier conserve 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":18747,"question":"Quel est le reste de la division de 2^10 par 11 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"2^10 = 1024. 1024 ÷ 11 = 93 avec un reste de 1, car 11 × 93 = 1023 et 1024 - 1023 = 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\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":18748,"question":"Si a et b sont deux entiers tels que a|b et b|a, alors a = b ou a = -b.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si a divise b et b divise a, alors |a| = |b|, donc a = b ou 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}]