Question

To determine congruence class \({(4)_8}\), where \(m\) is \(8\).

Short answer

The congruence class of \({(4)_8}\) is \(\{ \ldots , - 20, - 12, - 4,4,12,20,28,36, \ldots \} \).

Step-by-step solution

Given data

Given data is \(m = 8\).

Formula used of modulo

By the definition of modulo,

\(\begin{array}{c}{(4)_m} = \{ x \in Z\mid 4 \equiv x(\,\bmod \,m)\} \\ = \{ 4 + km:k \in Z\} \\ = \{ \ldots ,4 - 3m,4 - 2m,4 - m,4,4 + m,4 + 2m,4 + 3m,4 + 4m, \ldots \} \end{array}\).

Find congruence class

If \(m = 8\), then \({(4)_8}\) is,

\(\begin{array}{c}{(4)_8} = \{ \ldots ,4 - 3m,4 - 2m,4 - m,4,4 + m,4 + 2m,4 + 3m,4 + 4m, \ldots \} \\ = \{ \ldots ,4 - 3(8),4 - 2(8),4 - (8),4,4 + (8),4 + 2(8),4 + 3(8),4 + 4(8) \ldots \} \\ = \{ \ldots ,4 - 24,4 - 16,4 - 8,4,4 + 8,4 + 16,4 + 24,4 + 32, \ldots \} \\ = \{ \ldots , - 20, - 12, - 4,4,12,20,28,36, \ldots \} \end{array}\)

Hence, the congruence class of \({(4)_8}\) is \(\{ \ldots , - 20, - 12, - 4,4,12,20,28,36, \ldots \} \).