Colorful Power Sum

Find the last digit when

\[\color{blue}1^{\color{blue}1} + \color{green}2^{\color{green}2} + \color{purple}3^{ \color{purple}3} + \cdots + \color{red}9^{ \color{red}9} + \color{violet}{10}^{\color{violet}{10}} \]

is written out as an integer.

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