+166 What is the remainder when \(2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005\) is divided by 19?
+26393 What is the remainder when
\(2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005\)
is divided by 19?
\(\begin{array}{|rclcl|} \hline && 2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005 \pmod{19} \\ &&&& 2001 \pmod{19} \\ &&&\equiv& {\color{red}6} \pmod{19} \\\\ &&&& 2002 \pmod{19} \\ &&&\equiv& {\color{red}7} \pmod{19} \\\\ &&&& 2003 \pmod{19} \\ &&&\equiv& {\color{red}8} \pmod{19} \\\\ &&&& 2004 \pmod{19} \\ &&&\equiv& {\color{red}9} \pmod{19} \\\\ &&&& 2005 \pmod{19} \\ &&&\equiv& {\color{red}10} \pmod{19} \\\\ &\equiv& 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \pmod{19} \\ &\equiv& 30240 \pmod{19} \\ &\equiv& {\color{red}11} \pmod{19} \\ \hline \end{array}\)
The remainder is 11
