Colorful Power Sum

Find the last digit when

11+22+33++99+1010\color{#3D99F6}1^{\color{#3D99F6}1} + \color{#20A900}2^{\color{#20A900}2} + \color{#69047E}3^{ \color{#69047E}3} + \cdots + \color{#D61F06}9^{ \color{#D61F06}9} + \color{#BA33D6}{10}^{\color{#BA33D6}{10}}

is written out as an integer.

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