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What is the remainder when \(2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005\) is divided by 19?

 May 19, 2018
 #1
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(2001*2002*2003*2004*2005) mod 19 = 11

 May 19, 2018
 #2
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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

 

laugh

 May 22, 2018

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