Question

Assume that your computer has word size of $\mathbf{\text{100}}$. Use the method outlined in the text to find the sum of $\mathbf{123}\mathbf{,}\mathbf{684}\mathbf{+}\mathbf{413}\mathbf{,}\mathbf{456}$using, ${\mathit{m}}_{\mathbf{1}}\mathbf{=}\mathbf{95}\mathbf{,}{\mathit{m}}_{\mathbf{2}}\mathbf{=}\mathbf{97}\mathbf{,}{\mathit{m}}_{\mathbf{3}}\mathbf{=}\mathbf{98}\mathbf{,}{\mathit{m}}_{\mathbf{4}}\mathbf{=}\mathbf{99}$

Short answer

The sum of $123,684$and $413456$is obtained as: $537,140$.

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.

Considering the values

The Chinese remainder theorem is used:

The number$95,97,98,99$is$95\times 97\times 98\times 99=80,403,930$ is represented by using the reminder on division by module

The value of $123,684$is:

$$\begin{gathered} 123,684\left(\mathrm{mod}95\right); \\ 123,684\left(\mathrm{mod}97\right); \\ 123,684\left(\mathrm{mod}98\right); \\ 123,684\left(\mathrm{mod}99\right) \\ =\left(33,8,9,89\right) \end{gathered}$$

The value of $413456$is.

$$\begin{gathered} 413456\left(\mathrm{mod}95\right); \\ 413456\left(\mathrm{mod}97\right); \\ 413456\left(\mathrm{mod}98\right); \\ 413456\left(\mathrm{mod}99\right) \\ =\left(32,92,42,16\right) \end{gathered}$$

Step 3:- Adding the values

The value is obtained as:

$\left(33,8,9,89\right)+\left(32,92,42,16\right)=\left(65,2,5,1,10\right)$

The tuple is equals to $537,140$

Hence, the sum of $123,684$ and $413456$ is $537,140$.