sove-inside-Z-3Z-the-systeme-5x-7y-10-2x-5y-8-

Question Number 62439 by mathsolverby Abdo last updated on 21/Jun/19

s o v e i n s i d e Z / 3 Z t h e s y s t e m e

{\begin{cases} 5 x + 7 y = 10 \\ 2 x + 5 y = 8 \end{cases}

Commented by arcana last updated on 22/Jun/19

Z / 3 Z = Z_{3} = {\bar{0}, \bar{1}, \bar{2}}

\bar{0} = {\dots, - 9, - 6, - 3, 0, 3, 6, 9, \dots}

\bar{1} = {\dots, - 10, - 7, - 4, 1, 4, 7, 10, \dots}

\bar{2} = {\dots, - 11, - 8, - 5, 2, 5, 8, 11, \dots}

\bar{2} x + \bar{1} y = \bar{1}

\bar{2} x + \bar{2} y = \bar{2}

\Rightarrow \bar{2} y - y = \bar{2} - \bar{1} = \bar{1}

y = \bar{1} \Rightarrow \bar{2} x + (\bar{1}) = \bar{1}

\bar{2} x = \bar{0} (cancelativ) \Rightarrow x = \bar{0} cause (Z_{3}, \bar{+}, \bar{\cdot}) is a field

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