Set of Residue Classes/Examples

Examples of Sets of Residue Classes

Set of Residue Classes Modulo $2$

The elements of $\Z_2$, the set of residue classes modulo $2$, are:

\(\ds \eqclass 0 2\)

\(=\)

\(\ds \set {\dotsc, -6, -4, -2, 0, 2, 4, 6, \dotsc}\)

\(\ds \eqclass 1 2\)

\(=\)

\(\ds \set {\dotsc, -5, -3, -1, 1, 3, 5, \dotsc}\)

Set of Residue Classes Modulo $3$

The elements of $\Z_3$, the set of residue classes modulo $3$, are:

\(\ds \eqclass 0 3\)

\(=\)

\(\ds \set {\dotsc, -6, -3, 0, 3, 6, \dotsc}\)

\(\ds \eqclass 1 3\)

\(=\)

\(\ds \set {\dotsc, -5, -2, 1, 4, 7, \dotsc}\)

\(\ds \eqclass 2 3\)

\(=\)

\(\ds \set {\dotsc, -4, -1, 2, 5, 6, \dotsc}\)

Set of Residue Classes Modulo $4$

The elements of $\Z_4$, the set of residue classes modulo $4$, are:

\(\ds \eqclass 0 4\)

\(=\)

\(\ds \set {\dotsc, -8, -4, 0, 4, 8, 12, 16, \dotsc}\)

\(\ds \eqclass 1 4\)

\(=\)

\(\ds \set {\dotsc, -7, -3, 1, 5, 9, 13, 17, \dotsc}\)

\(\ds \eqclass 2 4\)

\(=\)

\(\ds \set {\dotsc, -6, -2, 2, 6, 10, 14, 18, \dotsc}\)

\(\ds \eqclass 3 4\)

\(=\)

\(\ds \set {\dotsc, -5, -1, 3, 7, 11, 15, 19, \dotsc}\)

Set of Residue Classes Modulo $5$

The elements of $\Z_5$, the set of residue classes modulo $5$, are:

\(\ds \eqclass 0 5\)

\(=\)

\(\ds \set {\dotsc, -10, -5, 0, 5, 10, 15, 20, \dotsc}\)

\(\ds \eqclass 1 5\)

\(=\)

\(\ds \set {\dotsc, -9, -4, 1, 6, 11, 16, 21, \dotsc}\)

\(\ds \eqclass 2 5\)

\(=\)

\(\ds \set {\dotsc, -8, -3, 2, 7, 12, 17, 22, \dotsc}\)

\(\ds \eqclass 3 5\)

\(=\)

\(\ds \set {\dotsc, -7, -2, 3, 8, 13, 18, 23, \dotsc}\)

\(\ds \eqclass 4 5\)

\(=\)

\(\ds \set {\dotsc, -6, -2, 1, 9, 14, 19, 24, \dotsc}\)