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What is the ones digit of 1^2009 + 2^2009 + 3^2009 + ... + 2009^2009 + 2010^2009 + 2011^2009 + 2012^2009?

 May 17, 2021
 #1
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The ones digits is 8.

 May 17, 2021
 #2
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What is the ones digit of
\(1^{2009} + 2^{2009} + 3^{2009} + \ldots + 2009^{2009} + 2010^{2009} + 2011^{2009} + 2012^{2009}\)?

 

\(\small{ \begin{array}{|rcll|} \hline && \mathbf{ 1^{2009} + 2^{2009} + 3^{2009} + 4^{2009} + 5^{2009} } \\ && \mathbf{ + 6^{2009} + 7^{2009} + 8^{2009} + 9^{2009} + 10^{2009} } \pmod {10} \\ && \boxed{ 6 \equiv -4\pmod {10} \\ 7 \equiv -3\pmod {10} \\ 8 \equiv -2\pmod {10} \\ 9 \equiv -1\pmod {10} \\ 10 \equiv 0\pmod {10} } \\ &=& 1^{2009} + 2^{2009} + 3^{2009} + 4^{2009} + 5^{2009} \\ && + (-4)^{2009} + (-3)^{2009} + (-2)^{2009} + (-1)^{2009} + 0^{2009} \pmod {10} \\ &=& 1^{2009} + 2^{2009} + 3^{2009} + 4^{2009} + 5^{2009} \\ && -4^{2009} -3^{2009} -2^{2009} -1^{2009} \pmod {10} \\ &=& \mathbf{ 5^{2009}\pmod {10} } \\ \hline && \mathbf{ 11^{2009} + 12^{2009} + 13^{2009} + 14^{2009} + 15^{2009} } \\ && \mathbf{ + 16^{2009} + 17^{2009} + 18^{2009} + 19^{2009} + 20^{2009} }\pmod {10} \\ &=& 1^{2009} + 2^{2009} + 3^{2009} + 4^{2009} + 5^{2009}\\ && -4^{2009} -3^{2009} -2^{2009} -1^{2009} \pmod {10} \\ &=& \mathbf{ 5^{2009}\pmod {10} } \\ && \ldots \\ \hline && 1^{2009} + 2^{2009} + 3^{2009} + \ldots \\ && + 2009^{2009} + 2010^{2009} \\ && + 2011^{2009} + 2012^{2009} \pmod {10} \\\\ &=& 1^{2009} + 2^{2009} + 3^{2009} + 4^{2009} + 5^{2009} \\ && + 6^{2009} + 7^{2009} + 8^{2009} + 9^{2009} + 10^{2009} \\ && + 11^{2009} + 12^{2009} + 13^{2009} + 14^{2009} + 15^{2009} \\ && + 16^{2009} + 17^{2009} + 18^{2009} + 19^{2009} + 20^{2009} \\ && \ldots \\ && + 2011^{2009} + 2012^{2009} + 2013^{2009} + 2014^{2009} + 2015^{2009} \\ && + 2016^{2009} + 2017^{2009} + 2018^{2009} + 2019^{2009} + 2020^{2009} \\ && + 2011^{2009} + 2012^{2009} \pmod {10} \\\\ &=& 201 * 5^{2009} + 2011^{2009} + 2012^{2009} \pmod {10} \\ && \boxed{ 5^{2009} \equiv 5\pmod {10} ~\text{circle } ~5,5,\ldots \\ 2011^{2009} \equiv 1^{2009} \equiv 1\pmod {10} \\ 2012^{2009} \equiv 2^{2009}\pmod {10} \equiv 2\pmod {10} ~\text{circle } ~2,4,8,6,2\ldots } \\ &=& 201 * 5 + 1 + 2 \pmod {10} \\ && \boxed{201 \equiv 1 \pmod{10} } \\ &=& 1*5 + 1 + 2 \\ &=& \mathbf{8}\pmod{10} \\ \hline \end{array} }\)

 

laugh

 May 17, 2021

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