# Endlicher Körper/25/Operationstafeln

Zur Navigation springen Zur Suche springen

Wir nehmen die Darstellung

${\displaystyle {\mathbb {F} }_{25}\cong {\mathbb {F} }_{5}[X]/(X^{2}+X+2)}$.

Dabei ist ${\displaystyle X^{2}+X+2}$ irreduzibel in ${\displaystyle {\mathbb {F} }_{5}[X]}$, da es keine Nullstelle in ${\displaystyle {\mathbb {F} }_{5}}$ besitzt. Es bezeichne ${\displaystyle x}$ die Restklasse von ${\displaystyle X}$.

 ${\displaystyle +}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 3}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x+1}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle x+2}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+3}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle x+4}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 2x}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle 3x}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 4x}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle x+4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$

 ${\displaystyle \cdot }$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 2}$ ${\displaystyle 0}$ ${\displaystyle 2}$ ${\displaystyle 4}$ ${\displaystyle 1}$ ${\displaystyle 3}$ ${\displaystyle 2x}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+3}$ ${\displaystyle 4x}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle x}$ ${\displaystyle x+2}$ ${\displaystyle x+4}$ ${\displaystyle x+1}$ ${\displaystyle x+3}$ ${\displaystyle 3x}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+3}$ ${\displaystyle 3}$ ${\displaystyle 0}$ ${\displaystyle 3}$ ${\displaystyle 1}$ ${\displaystyle 4}$ ${\displaystyle 2}$ ${\displaystyle 3x}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+1}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x+2}$ ${\displaystyle x}$ ${\displaystyle x+3}$ ${\displaystyle x+1}$ ${\displaystyle x+4}$ ${\displaystyle x+2}$ ${\displaystyle 4x}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x+2}$ ${\displaystyle 4}$ ${\displaystyle 0}$ ${\displaystyle 4}$ ${\displaystyle 3}$ ${\displaystyle 2}$ ${\displaystyle 1}$ ${\displaystyle 4x}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x+3}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+1}$ ${\displaystyle 3x}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+1}$ ${\displaystyle 2x}$ ${\displaystyle 2x+4}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+1}$ ${\displaystyle x}$ ${\displaystyle x+4}$ ${\displaystyle x+3}$ ${\displaystyle x+2}$ ${\displaystyle x+1}$ ${\displaystyle x}$ ${\displaystyle 0}$ ${\displaystyle x}$ ${\displaystyle 2x}$ ${\displaystyle 3x}$ ${\displaystyle 4x}$ ${\displaystyle 4x+3}$ ${\displaystyle 3}$ ${\displaystyle x+3}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+3}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+1}$ ${\displaystyle 1}$ ${\displaystyle x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x+4}$ ${\displaystyle 4}$ ${\displaystyle x+4}$ ${\displaystyle x+2}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 2}$ ${\displaystyle x+1}$ ${\displaystyle 0}$ ${\displaystyle x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 3}$ ${\displaystyle x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle 1}$ ${\displaystyle x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x}$ ${\displaystyle 4}$ ${\displaystyle x}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 2}$ ${\displaystyle x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x}$ ${\displaystyle 4x+1}$ ${\displaystyle x+2}$ ${\displaystyle 0}$ ${\displaystyle x+2}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle x+3}$ ${\displaystyle 2x}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+4}$ ${\displaystyle 1}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x}$ ${\displaystyle 2}$ ${\displaystyle x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x+1}$ ${\displaystyle 3}$ ${\displaystyle x}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 4}$ ${\displaystyle x+1}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x}$ ${\displaystyle x+3}$ ${\displaystyle 0}$ ${\displaystyle x+3}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+4}$ ${\displaystyle 2}$ ${\displaystyle x}$ ${\displaystyle 4x+1}$ ${\displaystyle 4}$ ${\displaystyle x+2}$ ${\displaystyle 2x}$ ${\displaystyle 3x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x}$ ${\displaystyle 4x+3}$ ${\displaystyle 1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x}$ ${\displaystyle 3}$ ${\displaystyle x+1}$ ${\displaystyle 2x+4}$ ${\displaystyle x+4}$ ${\displaystyle 0}$ ${\displaystyle x+4}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+1}$ ${\displaystyle 3x+3}$ ${\displaystyle 4x+2}$ ${\displaystyle 1}$ ${\displaystyle x}$ ${\displaystyle 2x+4}$ ${\displaystyle x+1}$ ${\displaystyle 2x}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x+3}$ ${\displaystyle 2}$ ${\displaystyle 4x+4}$ ${\displaystyle 3}$ ${\displaystyle x+2}$ ${\displaystyle 2x+1}$ ${\displaystyle 3x}$ ${\displaystyle 2x+2}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x}$ ${\displaystyle 4}$ ${\displaystyle x+3}$ ${\displaystyle 2x}$ ${\displaystyle 0}$ ${\displaystyle 2x}$ ${\displaystyle 4x}$ ${\displaystyle x}$ ${\displaystyle 3x}$ ${\displaystyle 3x+1}$ ${\displaystyle 1}$ ${\displaystyle 2x+1}$ ${\displaystyle 4x+1}$ ${\displaystyle x+1}$ ${\displaystyle x+2}$ ${\displaystyle 3x+2}$ ${\displaystyle 2}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle x+3}$ ${\displaystyle 3x+3}$ ${\displaystyle 3}$ ${\displaystyle 2x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 4x+4}$ ${\displaystyle x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle 4}$ ${\displaystyle 2x+1}$ ${\displaystyle 0}$ ${\displaystyle 2x+1}$ ${\displaystyle 4x+2}$ ${\displaystyle x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 4x+1}$ ${\displaystyle x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle 4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+2}$ ${\displaystyle 3}$ ${\displaystyle 2x+4}$ ${\displaystyle 4x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle x}$ ${\displaystyle 3x+1}$ ${\displaystyle 2}$ ${\displaystyle x+4}$ ${\displaystyle 3x}$ ${\displaystyle 1}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x+2}$ ${\displaystyle 0}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x+4}$ ${\displaystyle x+1}$ ${\displaystyle 3x+3}$ ${\displaystyle 1}$ ${\displaystyle 2x+3}$ ${\displaystyle 4x}$ ${\displaystyle x+2}$ ${\displaystyle 3x+4}$ ${\displaystyle 2}$ ${\displaystyle 2x+4}$ ${\displaystyle 4x+1}$ ${\displaystyle x+3}$ ${\displaystyle 3x}$ ${\displaystyle 3}$ ${\displaystyle 2x}$ ${\displaystyle 4x+2}$ ${\displaystyle x+4}$ ${\displaystyle 3x+1}$ ${\displaystyle 4}$ ${\displaystyle 2x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle x}$ ${\displaystyle 3x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 0}$ ${\displaystyle 2x+3}$ ${\displaystyle 4x+1}$ ${\displaystyle x+4}$ ${\displaystyle 3x+2}$ ${\displaystyle x+1}$ ${\displaystyle 3x+4}$ ${\displaystyle 2}$ ${\displaystyle 2x}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x}$ ${\displaystyle x+3}$ ${\displaystyle 3x+1}$ ${\displaystyle 4}$ ${\displaystyle 3x+3}$ ${\displaystyle 1}$ ${\displaystyle 2x+4}$ ${\displaystyle 4x+2}$ ${\displaystyle x}$ ${\displaystyle 4x+4}$ ${\displaystyle x+2}$ ${\displaystyle 3x}$ ${\displaystyle 3}$ ${\displaystyle 2x+1}$ ${\displaystyle 2x+4}$ ${\displaystyle 0}$ ${\displaystyle 2x+4}$ ${\displaystyle 4x+3}$ ${\displaystyle x+2}$ ${\displaystyle 3x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle 4x}$ ${\displaystyle x+4}$ ${\displaystyle 3x+3}$ ${\displaystyle 2}$ ${\displaystyle 4x+2}$ ${\displaystyle x+1}$ ${\displaystyle 3x}$ ${\displaystyle 4}$ ${\displaystyle 2x+3}$ ${\displaystyle x+3}$ ${\displaystyle 3x+2}$ ${\displaystyle 1}$ ${\displaystyle 2x}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle 3}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x+1}$ ${\displaystyle x}$ ${\displaystyle 3x}$ ${\displaystyle 0}$ ${\displaystyle 3x}$ ${\displaystyle x}$ ${\displaystyle 4x}$ ${\displaystyle 2x}$ ${\displaystyle 2x+4}$ ${\displaystyle 4}$ ${\displaystyle 3x+4}$ ${\displaystyle x+4}$ ${\displaystyle 4x+4}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x+3}$ ${\displaystyle 3}$ ${\displaystyle 3x+3}$ ${\displaystyle x+3}$ ${\displaystyle x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x+2}$ ${\displaystyle 2}$ ${\displaystyle 3x+2}$ ${\displaystyle 3x+1}$ ${\displaystyle x+1}$ ${\displaystyle 4x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle 1}$ ${\displaystyle 3x+1}$ ${\displaystyle 0}$ ${\displaystyle 3x+1}$ ${\displaystyle x+2}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle x}$ ${\displaystyle 4x+1}$ ${\displaystyle 2x+2}$ ${\displaystyle 3}$ ${\displaystyle x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 2x}$ ${\displaystyle 1}$ ${\displaystyle 3x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle 4}$ ${\displaystyle 3x}$ ${\displaystyle x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle 2}$ ${\displaystyle 3x+3}$ ${\displaystyle x+4}$ ${\displaystyle 4x}$ ${\displaystyle 3x+2}$ ${\displaystyle 0}$ ${\displaystyle 3x+2}$ ${\displaystyle x+4}$ ${\displaystyle 4x+1}$ ${\displaystyle 2x+3}$ ${\displaystyle 4x+4}$ ${\displaystyle 2x+1}$ ${\displaystyle 3}$ ${\displaystyle 3x}$ ${\displaystyle x+2}$ ${\displaystyle 3x+3}$ ${\displaystyle x}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x+4}$ ${\displaystyle 1}$ ${\displaystyle 2x+2}$ ${\displaystyle 4}$ ${\displaystyle 3x+1}$ ${\displaystyle x+3}$ ${\displaystyle 4x}$ ${\displaystyle x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x}$ ${\displaystyle 2}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x+3}$ ${\displaystyle 0}$ ${\displaystyle 3x+3}$ ${\displaystyle x+1}$ ${\displaystyle 4x+4}$ ${\displaystyle 2x+2}$ ${\displaystyle 4}$ ${\displaystyle 3x+2}$ ${\displaystyle x}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x+1}$ ${\displaystyle 3}$ ${\displaystyle 3x+1}$ ${\displaystyle x+4}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x}$ ${\displaystyle 2}$ ${\displaystyle 3x}$ ${\displaystyle x+3}$ ${\displaystyle 4x+1}$ ${\displaystyle 2x+4}$ ${\displaystyle 1}$ ${\displaystyle 3x+4}$ ${\displaystyle x+2}$ ${\displaystyle 4x}$ ${\displaystyle 2x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 0}$ ${\displaystyle 3x+4}$ ${\displaystyle x+3}$ ${\displaystyle 4x+2}$ ${\displaystyle 2x+1}$ ${\displaystyle x+4}$ ${\displaystyle 4x+3}$ ${\displaystyle 2x+2}$ ${\displaystyle 1}$ ${\displaystyle 3x}$ ${\displaystyle 2x+3}$ ${\displaystyle 2}$ ${\displaystyle 3x+1}$ ${\displaystyle x}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x+2}$ ${\displaystyle x+1}$ ${\displaystyle 4x}$ ${\displaystyle 2x+4}$ ${\displaystyle 3}$ ${\displaystyle 4x+1}$ ${\displaystyle 2x}$ ${\displaystyle 4}$ ${\displaystyle 3x+3}$ ${\displaystyle x+2}$ ${\displaystyle 4x}$ ${\displaystyle 0}$ ${\displaystyle 4x}$ ${\displaystyle 3x}$ ${\displaystyle 2x}$ ${\displaystyle x}$ ${\displaystyle x+2}$ ${\displaystyle 2}$ ${\displaystyle 4x+2}$ ${\displaystyle 3x+2}$ ${\displaystyle 2x+2}$ ${\displaystyle 2x+4}$ ${\displaystyle x+4}$ ${\displaystyle 4}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x+4}$ ${\displaystyle 3x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle x+1}$ ${\displaystyle 1}$ ${\displaystyle 4x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle 3x+3}$ ${\displaystyle 2x+3}$ ${\displaystyle x+3}$ ${\displaystyle 3}$ ${\displaystyle 4x+1}$ ${\displaystyle 0}$ ${\displaystyle 4x+1}$ ${\displaystyle 3x+2}$ ${\displaystyle 2x+3}$ ${\displaystyle x+4}$ ${\displaystyle 2x+2}$ ${\displaystyle x+3}$ ${\displaystyle 4}$ ${\displaystyle 4x}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x}$ ${\displaystyle 2x+1}$ ${\displaystyle x+2}$ ${\displaystyle 3}$ ${\displaystyle x+1}$ ${\displaystyle 2}$ ${\displaystyle 4x+3}$ ${\displaystyle 3x+4}$ ${\displaystyle 2x}$ ${\displaystyle 3x+3}$ ${\displaystyle 2x+4}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 4x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 0}$ ${\displaystyle 4x+2}$ ${\displaystyle 3x+4}$ ${\displaystyle 2x+1}$ ${\displaystyle x+3}$ ${\displaystyle 3x+2}$ ${\displaystyle 2x+4}$ ${\displaystyle x+1}$ ${\displaystyle 3}$ ${\displaystyle 4x}$ ${\displaystyle x+4}$ ${\displaystyle 1}$ ${\displaystyle 4x+3}$ ${\displaystyle 3x}$ ${\displaystyle 2x+2}$ ${\displaystyle 4x+1}$ ${\displaystyle 3x+3}$ ${\displaystyle 2x}$ ${\displaystyle x+2}$ ${\displaystyle 4}$ ${\displaystyle 2x+3}$ ${\displaystyle x}$ ${\displaystyle 2}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x+1}$ ${\displaystyle 4x+3}$ ${\displaystyle 0}$ ${\displaystyle 4x+3}$ ${\displaystyle 3x+1}$ ${\displaystyle 2x+4}$ ${\displaystyle x+2}$ ${\displaystyle 4x+2}$ ${\displaystyle 3x}$ ${\displaystyle 2x+3}$ ${\displaystyle x+1}$ ${\displaystyle 4}$ ${\displaystyle 3x+4}$ ${\displaystyle 2x+2}$ ${\displaystyle x}$ ${\displaystyle 3}$ ${\displaystyle 4x+1}$ ${\displaystyle 2x+1}$ ${\displaystyle x+4}$ ${\displaystyle 2}$ ${\displaystyle 4x}$ ${\displaystyle 3x+3}$ ${\displaystyle x+3}$ ${\displaystyle 1}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x+2}$ ${\displaystyle 2x}$ ${\displaystyle 4x+4}$ ${\displaystyle 0}$ ${\displaystyle 4x+4}$ ${\displaystyle 3x+3}$ ${\displaystyle 2x+2}$ ${\displaystyle x+1}$ ${\displaystyle 2}$ ${\displaystyle 4x+1}$ ${\displaystyle 3x}$ ${\displaystyle 2x+4}$ ${\displaystyle x+3}$ ${\displaystyle 4}$ ${\displaystyle 4x+3}$ ${\displaystyle 3x+2}$ ${\displaystyle 2x+1}$ ${\displaystyle x}$ ${\displaystyle 1}$ ${\displaystyle 4x}$ ${\displaystyle 3x+4}$ ${\displaystyle 2x+3}$ ${\displaystyle x+2}$ ${\displaystyle 3}$ ${\displaystyle 4x+2}$ ${\displaystyle 3x+1}$ ${\displaystyle 2x}$ ${\displaystyle x+4}$