Arithmétique dans ℤ : Division euclidienne et congruences

Question 1 sur 10 10:00

[{"id":49941,"question":"Quel est le reste de la division euclidienne de -17 par 5 ?","option_a":"-2","option_b":"2","option_c":"3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"C","explication":"La division euclidienne de -17 par 5 donne un quotient de -4 et un reste de 3, car -17 = 5 × (-4) + 3 et 0 ≤ 3 \u003C 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49942,"question":"Si a ≡ b [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 [n] signifie que n divise (a - b).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49943,"question":"Quel est le quotient de la division euclidienne de 45 par -7 ?","option_a":"-6","option_b":"-7","option_c":"6","option_d":"7","option_e":"","option_f":"","bonne_reponse":"A","explication":"45 = (-7) × (-6) + 3, donc le quotient est -6 et le reste est 3 (0 ≤ 3 \u003C 7).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49944,"question":"Si 12 ≡ x [5], alors x peut être égal à :","option_a":"2","option_b":"7","option_c":"12","option_d":"17","option_e":"","option_f":"","bonne_reponse":"B","explication":"12 ≡ 7 [5] car 12 - 7 = 5, qui est divisible par 5. Les autres options ne satisfont pas cette condition.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49945,"question":"La somme de deux entiers pairs est toujours paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Soient a = 2k et b = 2m, alors a + b = 2(k + m), qui est un entier pair.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49946,"question":"Quel est le reste de la division euclidienne de 100 par 7 ?","option_a":"2","option_b":"3","option_c":"4","option_d":"5","option_e":"","option_f":"","bonne_reponse":"B","explication":"100 = 7 × 14 + 2, donc le reste est 2 (0 ≤ 2 \u003C 7).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49947,"question":"Si a ≡ 3 [4] et b ≡ 2 [4], alors a + b ≡ ? [4]","option_a":"0","option_b":"1","option_c":"5","option_d":"6","option_e":"","option_f":"","bonne_reponse":"D","explication":"a + b ≡ (3 + 2) [4] ≡ 5 [4]. Comme 5 ≡ 1 [4], le résultat est 1, mais 5 est aussi correct dans le contexte de la congruence.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49948,"question":"Le produit de deux entiers impairs est toujours impair.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Soient a = 2k + 1 et b = 2m + 1, alors a × b = 4km + 2k + 2m + 1 = 2(2km + k + m) + 1, qui est impair.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49949,"question":"Quel est le plus petit entier positif congru à -8 modulo 5 ?","option_a":"2","option_b":"3","option_c":"4","option_d":"7","option_e":"","option_f":"","bonne_reponse":"A","explication":"-8 ≡ 2 [5] car -8 + 10 = 2, et 2 est le plus petit entier positif satisfaisant cette congruence.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":49950,"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":"En élevant au carré les deux côtés de a ≡ b [n], on obtient a² ≡ b² [n] car (a - b) est divisible par n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]