# Number Theory/Residue class

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Residue classes are sets with specific properties.

## Definition: residue class

Let ${\displaystyle a \in \mathbb{Z}}$ and ${\displaystyle c \in \mathbb{N}}$ be fixed. Then we call the set:

${\displaystyle [a] := \{b \in \mathbb{Z} \vert~b \equiv a ~(\hspace{-10pt}\mod c) \} = \{ b~(\hspace{-10pt}\mod c) = a~(\hspace{-10pt}\mod c) \} }$

a residue class.

### Example

Let ${\displaystyle n \in \mathbb{N}}$. Then we have:

${\displaystyle \begin{array}{lllll} n \geq 0: 3 \cdot 7 & + 30 \cdot n & \Longleftrightarrow & [b_1] \equiv 3 \cdot 7 & ~(\hspace{-7pt}\mod 30) \\ & & & & \\ n \geq 1: 3 & + 30 \cdot n & \Longleftrightarrow & [b_2] \equiv 3 & ~(\hspace{-7pt}\mod 30)\\ & & & & \\ n \geq 0: 3 \cdot 9 & +30 \cdot n & \Longleftrightarrow & [b_3] \equiv 3 \cdot 9 & ~(\hspace{-7pt}\mod 30)\\ & & & & \\ n \geq 0: 3 \cdot 3 & + 30 \cdot n & \Longleftrightarrow & [b_4] \equiv 3 \cdot 3 & ~(\hspace{-7pt}\mod 30)\\ & & & & \\ n \geq 0: 7 \cdot 13 & + 70 \cdot n & \Longleftrightarrow & [b_5] \equiv 7 \cdot 13 & ~(\hspace{-7pt}\mod 70) \\ & & & & \\ n \geq 1: 7 \cdot 9 & + 70 \cdot n & \Longleftrightarrow & [b_6] \equiv 7 \cdot 9 & ~(\hspace{-7pt}\mod 70) \\ & & & & \\ n \geq 0: 7 \cdot 11 & + 70 \cdot n & \Longleftrightarrow & [b_7] \equiv 7 \cdot 11 & ~(\hspace{-7pt}\mod 70) \\ & & & & \\ n \geq 0: 7 \cdot 7 & + 70 \cdot n & \Longleftrightarrow & [b_8] \equiv 7 \cdot 7 & ~(\hspace{-7pt}\mod 70) \\ & & & & \\ \hspace{35pt} \vdots & \hspace{15pt} \vdots & \hspace{5pt} \vdots & \hspace{35pt} \vdots & \hspace{25pt} \vdots \\ \end{array}}$

## Definition: combinatorial residue class

Let ${\displaystyle p_{\sigma t}}$, ${\displaystyle p_{\zeta \tau} \in P_{\sigma t}}$ be fixed, whereby ${\displaystyle P_{\sigma t}}$ denotes the prime number matrix, and ${\displaystyle \delta \in \mathbb{N}}$. Then we call the set:

${\displaystyle [b_i] := p_{\zeta \tau} p_{\sigma t}~(\hspace{-9pt}\mod \delta) }$

a combinatorial residue class.