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[A-L] (2023/24) Foglio 3 - Esercizio 3 #184

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  • (1) $897$ e $4403$ sono coprimi se $(897,4403)=1$
    Usando l'algoritmo di Euclide:
    $4403=897\cdot 4 +815$
    $897=815\cdot1+82$
    $815=82\cdot9+ 77$
    $82=77\cdot 1 +5$
    $77=5\cdot 15 + 2$
    $5=2\cdot 2+ 1$
    $2=1\cdot 2 + 0$
    $\Rightarrow (897,4403)=1$

  • (2) Cerchiamo un'identità di Bezout:
    $815 = a-b\cdot 4=(1,0)+(0,-4)=(1,-4)$
    $82=b-815\cdot 1=(0,1)+(1,-4)(-1)=(-1,5)$
    $77=815-82\cdot 9=(1,-4)+(-1,5)(-9)=(10,-49)$
    $5=82-77\cdot 1=(-1,5)+(10,-49)(-1)=(-11,54)$
    $2=77-5\cdot 15=(10,-49)+(-11,54)(-15)=(175,-859)$
    $1= 5-2\cdot 2=(-11,54)+(175,-859)(-2)=(-361,1772)$
    $\Rightarrow (\tilde{x},\tilde{y})=(1772,-361)$

  • (3) Vogliamo verificare che $897(\tilde{x}+x_0)+4403(\tilde{y}+y_0)=1$
    Per associatività $8…

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[A-L] (2023/24) Foglio 3 Esercizi tratti dal 3° foglio di esercizi (21 ottobre) dei Proff. Piazza e Viaggi (A.A. 2023/24)
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Converted from issue

This discussion was converted from issue #36 on December 15, 2023 12:17.