[A-L] (2023/24) Foglio 3 - Esercizio 3 #184
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Analogo a @Elia-Belli tranne punto 4: |
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Soluzione ad opera della tutor Elisa Scandiuzzi, lezione del 23 ottobre 2024 ( |
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(1)$897$ e $4403$ sono coprimi se $(897,4403)=1$
$4403=897\cdot 4 +815$
$897=815\cdot1+82$
$815=82\cdot9+ 77$
$82=77\cdot 1 +5$
$77=5\cdot 15 + 2$
$5=2\cdot 2+ 1$
$2=1\cdot 2 + 0$
$\Rightarrow (897,4403)=1$
Usando l'algoritmo di Euclide:
(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$897(\tilde{x}+x_0)+4403(\tilde{y}+y_0)=1$ $8…
Per associatività