Question

To determine congruence class \({(n)_5}\), where \(n\) is \( - 3\).

Short answer

The equivalence class is \(\{ \ldots , - 8, - 3,2,7,12, \ldots \} \).

Step-by-step solution

Given data

Given data is \(n = - 3\).

Formula used of modulo

By the definition of modulo,

\({(n)_5} = \{ i\mid i \equiv n(\,\bmod \,5)\} \).

Find equivalence class

Let \(i\) be the equivalence class of \( - 3\) contains all integers such that \(i \equiv - 3(\,\bmod \,5)\).

In this class, there are those integers that have a remainder of \(2(5 - 3 = 2)\) when we divide it by 5 .

\(\begin{array}{c}{( - 3)_5} = \{ i\mid i\\ \equiv - 3(\,\bmod \,5)\} \\ = \{ - 3 + 5c\mid c{\rm{ is aninteger }}\} \\ = \{ 2 - 5 + 5c\mid c{\rm{ is aninteger }}\} \\ = \{ 2 + 5d\mid d{\rm{ is aninteger }}\} \\ = \{ \ldots , - 8, - 3,2,7,12, \ldots \} \end{array}\)

Hence, the equivalence class is \(\{ \ldots , - 8, - 3,2,7,12, \ldots \} \).