Question

In exercise 8-13, solve the system of congruences

Short answer

The system of congruences is -30

Step-by-step solution

Conceptual Introduction

A theorem known as the Chinese remainder theorem provides a singular answer to simultaneous linear congruences with coprime moduli.

Evaluating the system of congruences

It is evident by the equations that:

$$\begin{gathered} \text{x}\equiv \text{3(mod11)} \\ \text{x}\equiv \text{4(mod17)} \end{gathered}$$

Where, $m=11,n=17,a=3,b=4$

By using $mu+nv=1$, find the values of u and v as,

$\left(17\right)\left(2\right)+\left(-3\right)\left(11\right)=1$

Then, $\text{u}=-\text{3,v}=\text{2}$.

The equation is then evaluated using the Chinese remainder theorem as:

$\begin{array}{rcl}x& =& bmu+anv\\ & =& \left(4\right)\left(11\right)(-3)+\left(3\right)\left(17\right)\left(2\right)\\ & =& -132+102\\ & =& -30\\ & & \end{array}$

It satisfies the given system of equations.

Therefore, the value obtained is -30