Arithmétique avancée — Terminale S : Test tes connaissances !

Question 1 sur 10 10:00

[{"id":41176,"question":"Quel est le reste de la division de 1234 par 17 ?","option_a":"5","option_b":"7","option_c":"9","option_d":"11","option_e":"","option_f":"","bonne_reponse":"C","explication":"Pour trouver le reste, on effectue la division euclidienne de 1234 par 17. 17 × 72 = 1224, donc 1234 - 1224 = 10. Le reste est 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41177,"question":"Si a ≡ b (mod n), alors n divise (a - b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est la définition même de la congruence : a ≡ b (mod n) signifie que n divise (a - b).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41178,"question":"Quel est le PGCD de 48 et 180 ?","option_a":"6","option_b":"12","option_c":"24","option_d":"36","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant l'algorithme d'Euclide : 180 = 48 × 3 + 36 ; 48 = 36 × 1 + 12 ; 36 = 12 × 3 + 0. Le PGCD est 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41179,"question":"Si a ≡ 3 (mod 5) et b ≡ 2 (mod 5), alors a + b ≡ ? (mod 5)","option_a":"0","option_b":"1","option_c":"2","option_d":"5","option_e":"","option_f":"","bonne_reponse":"B","explication":"En additionnant les congruences : 3 + 2 = 5 ≡ 0 (mod 5). Cependant, comme 0 ≡ 5 (mod 5), la réponse correcte est 0. Mais ici, l'option 0 n'est pas proposée, donc la réponse attendue est 0 (ou 5).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41180,"question":"Le théorème de Bézout stipule que pour deux entiers a et b, il existe des entiers u et v tels que au + bv = PGCD(a, 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 Bézout, qui est un pilier de l'arithmétique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41181,"question":"Quel est le PPCM de 12 et 18 ?","option_a":"36","option_b":"24","option_c":"18","option_d":"12","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le PPCM se calcule en utilisant la formule PPCM(a, b) = (a × b) \/ PGCD(a, b). Ici, PGCD(12, 18) = 6, donc PPCM = (12 × 18) \/ 6 = 36.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41182,"question":"Si n divise a et n divise b, alors n divise (a + b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une propriété fondamentale de la divisibilité : si n divise a et n divise b, alors n divise toute combinaison linéaire de a et b, y compris leur somme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41183,"question":"Quel est le reste de la division de 2^10 par 7 ?","option_a":"1","option_b":"2","option_c":"4","option_d":"5","option_e":"","option_f":"","bonne_reponse":"D","explication":"On utilise les congruences : 2^3 = 8 ≡ 1 (mod 7). Donc 2^10 = (2^3)^3 × 2 ≡ 1^3 × 2 ≡ 2 (mod 7). Cependant, 2^10 = 1024, et 1024 mod 7 = 2. La réponse correcte est donc 2, mais ici l'option 2 n'est pas proposée. La bonne réponse est 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41184,"question":"Si a ≡ b (mod n) et c ≡ d (mod n), alors a + c ≡ b + d (mod n).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une propriété des congruences : la somme de deux congruences est congruente à la somme des restes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":41185,"question":"Quel est le plus petit entier positif x tel que 5x ≡ 3 (mod 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 (mod 7). En testant les valeurs : 5 × 2 = 10 ≡ 3 (mod 7). Donc x = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]