Quiz : Arithmétique dans ℤ — Niveau Bac Sciences

Question 1 sur 10 10:00

[{"id":70932,"question":"Quel est le PGCD de 48 et 18 ?","option_a":"6","option_b":"12","option_c":"18","option_d":"24","option_e":"","option_f":"","bonne_reponse":"A","explication":"En appliquant l'algorithme d'Euclide : 48 = 2×18 + 12, puis 18 = 1×12 + 6, et enfin 12 = 2×6 + 0. Le PGCD est le dernier reste non nul, soit 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70933,"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":"C'est une propriété fondamentale des congruences : la somme de deux congruences modulo n est congruente à la somme des restes modulo n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70934,"question":"Quel est le PPCM de 12 et 15 ?","option_a":"30","option_b":"60","option_c":"90","option_d":"120","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le PPCM se calcule en multipliant les facteurs premiers avec les exposants les plus élevés : 12 = 2²×3 et 15 = 3×5. Donc PPCM = 2²×3×5 = 60.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70935,"question":"Si a divise b et b divise c, alors a divise c.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une propriété transitive de la divisibilité : si a divise b et b divise c, alors a divise c.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70936,"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":"A","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},{"id":70937,"question":"Si a et b sont premiers entre eux, alors PGCD(a, b) = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, deux entiers sont premiers entre eux si leur PGCD est égal à 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70938,"question":"Quel est le nombre de diviseurs de 36 ?","option_a":"6","option_b":"8","option_c":"9","option_d":"12","option_e":"","option_f":"","bonne_reponse":"B","explication":"La décomposition en facteurs premiers de 36 est 2²×3². Le nombre de diviseurs est (2+1)×(2+1) = 9.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70939,"question":"Si a ≡ 3 [5] et b ≡ 2 [5], alors a×b ≡ 6 [5].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"a×b ≡ 3×2 ≡ 6 ≡ 1 [5], car 6 mod 5 = 1. L'affirmation est donc fausse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70940,"question":"Quel est le plus petit entier naturel non nul divisible par 4 et 6 ?","option_a":"12","option_b":"24","option_c":"36","option_d":"48","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le PPCM de 4 et 6 est 12, qui est le plus petit entier divisible par les deux nombres.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":70941,"question":"Si a ≡ b [n], alors a² ≡ b² [n].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une conséquence directe des propriétés des congruences : si a ≡ b [n], alors a² ≡ b² [n].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]