# Interesting Summation

Algebra Level 5

When $$M$$ is a digit, $M + \overline{MM} + \overline{MMM} + \cdots + \underbrace{\overline{MMM\cdots M}}_{n \ M\text{'s}} = \frac{M}{a}\left [ 10^{n+1}-10-9n \right ].$ Find the value of $$a.$$

Note that $$\overline{MM}$$ represents a two-digit number with both digits being $$M.$$

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