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dw20407 - Matematica Discreta II Esame del Esercizio 1...

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Matematica Discreta II Esame del 21-07-2004 Esercizio 1. Determinare tutte le soluzione dell’equazione 238 x + 646 y = 306, con x, y ZZ . (3 pt) Esercizio 2. Consideriamo la ricorrenza a n = - a n - 1 + 2 a n - 2 + 6 n - 4 per n 2. a.) Dimostrare che a n = n 2 + n + 1, n 0, ` e una soluzione della ricorrenza. (1 pt) b.) Trovare tutti le soluzioni della ricorrenza. (3 pt) c.) Trovare la soluzione con a 0 = 2 e a 1 = - 2. (1 pt) Esercizio 3. a.) Quanti numeri di 9 cifre si pu` o fare usando le cifre di 112112119. (2 pt) b.) Quanti numeri di 9 cifre si pu` o fare, usando le cifre di 112112119, che sono divisibile per 11. (2 pt) Esercizio 4. Trovare il numero dei x ZZ con 1 x 1122345 che a.) sono divisibile per 7, 12 o 55. (3 pt) b.) sono divisibile per 15 e divisible per almeno due dei numeri 7, 12, 55. (1 pt) Esercizio 5. Quante soluzioni ci sono dell’equazione x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 = 45, dove x 1 , . . . , x 8 ZZ e x 1 , . . . , x 8 0, con a.) x 1 < 10. (3 pt) b.) x 1 + x 3 + x 5 + x 7 = 9 e x 3 = x 5 . (2 pt) Esercizio 6.
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