Question

In exercise 8-13, solve the system of congruences

$$\begin{gathered} \text{x}\equiv \text{5(mod6)} \\ \text{x}\equiv \text{7(mod11)} \end{gathered}$$

Short answer

The system of congruences is: 29

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:

$$\begin{gathered} x\equiv 5\left(\mathrm{mod}6\right) \\ x\equiv 7\left(\mathrm{mod}11\right) \end{gathered}$$

Where, $m=6,n=11,a=5,b=7$

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

$\left(6\right)\left(2\right)+\left(-1\right)\left(11\right)=1$

Then, $\text{u}=\text{2,v}=-\text{1}$

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

$\begin{array}{rcl}\text{x}& =& bmu+anv\\ & =& \left(7\right)\left(6\right)\left(2\right)+\left(5\right)\left(11\right)(-1)\\ & =& 84-55\\ & =& 29\\ & & \end{array}$

It satisfies the given system of equations.

Therefore, the value obtained is: 29 .