System of equations with no solutions

Algebra Level 5

For what integer \(0 < n < 1000 \), does the following system of n linear equations have no complex solution? \[ \begin{array}{ l l l l l l } -10a_1 & + a_2 & + a_3 & + \ldots & + a_n &= 9 \\ a_1 & - 10a_2 &+ a_3 & + \ldots & + a_n &= 99 \\ a_1 & + a_2 & - 10 a_3 & + \ldots & + a_n &= 999 \\ \vdots & & & & &\vdots \\ a_1 & + a_2 & + a_3 & + \ldots & - 10 a_n &= 10^n-1 \\ \end{array}\]

Details and assumptions

Clarification: If the system always admits a solution, type your answer as 0. If there are multiple cases where the system doesn't admit a solution, type in any one of your answers.

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