Informaţii, definiţii, teoreme, formule, exerciţii şi probleme rezolvate din matematica de liceu. RSS/XML

Data publicarii: 09 Iulie, 2010

EXEMPLUL 1

Suport teoretic:

Inelul claselor de resturi modulo 6, divizori ai lui zero.

Enunt:

Sa se rezolve sistemul urmator in inelul claselor de resturi modulo 6:

$\begin{cases}\hat{2}x+\hat{2}y=\hat{4}\\\hat{4}x+y=\hat{2}\end{cases}.$ \begin{cases}\hat{2}x+\hat{2}y=\hat{4}\\\hat{4}x+y=\hat{2}\end{cases}.

Raspuns:

$\mathcal{S}=\{(\hat{0},\hat{2}),(\hat{1},\hat{4}),(\hat{2},\hat{0}),(\hat{3},\hat{2}),(\hat{4},\hat{4}),(\hat{5},\hat{0})\}.$ \mathcal{S}=\{(\hat{0},\hat{2}),(\hat{1},\hat{4}),(\hat{2},\hat{0}),(\hat{3},\hat{2}),(\hat{4},\hat{4}),(\hat{5},\hat{0})\}.

Rezolvare:

Folosind metoda substitutiei, obtinem succesiv:

${\hat{2}x+{\hat{2}}\cdot{(\hat{2}-\hat{4}x)}=\hat{4}}\Leftrightarrow{\hat{2}x+\hat{4}-\hat{8}x=\hat{4}}\Leftrightarrow{(-\hat{6})x=\hat{0}}\Leftrightarrow{\hat{0}\cdot{x}=\hat{0}}.$ {\hat{2}x+{\hat{2}}\cdot{(\hat{2}-\hat{4}x)}=\hat{4}}\Leftrightarrow{\hat{2}x+\hat{4}-\hat{8}x=\hat{4}}\Leftrightarrow{(-\hat{6})x=\hat{0}}\Leftrightarrow{\hat{0}\cdot{x}=\hat{0}}.