Exercise 2.2.13 (Example of Problem 12)

Question

Solve $3x \equiv 1 \pmod {125}$ by Problem 12, taking $x_1 = 2$.

Richard Ganaye · Accepted Answer

\begin{proof} By Exercise 12, the solution of $3 x \equiv 1 \pmod{5^3}$ is $x_s$ the nearest integer to $y_s= -\frac{(1-ax_1)^s}{a}$.

Here $x_1 = 2, a = 3, s = 3$, thus 
$$y_s = - \frac{1}{3} (1 - 3 	imes 2)^3 = \frac{5^3}{3} = \frac{125}{3} = 42 - \frac{1}{3}.$$ 
Therefore the solution of $3x \equiv 1 \pmod {125}$ is $x_s = 42$.

Check: $3 	imes 42 = 126 \equiv 1 \pmod {125}$.
\end{proof}

Exercise 2.2.13 (Example of Problem 12)

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Exercise 2.2.13 (Example of Problem 12)

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