Interesting Summation

Algebra Level 4

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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