Answer to Question #314866 in Abstract Algebra for Jyo

Question

Solve the equation 3𝑥 = 2 𝑖𝑛 𝑍7 𝑎𝑛𝑑 𝑖𝑛 𝑍23

Answer 1

$i:\\7=2\cdot 3+1\\1=7-2\cdot 3\\\Rightarrow 3^{-1}=-2mod7=5mod7\Rightarrow x=3^{-1}\cdot 2=5\cdot 2mod7=3mod7\\ii:\\23=7\cdot 3+2\\3=2+1\\1=3-2=3-\left( 23-7\cdot 3 \right) =-23+8\cdot 3\Rightarrow \\\Rightarrow 3^{-1}=8mod23\Rightarrow x=2\cdot 8mod23=16mod23$