How Many 8's?

Let \(x_n\) be defined as the smallest positive integer satisfying \[\large x_n ^3 \equiv \underbrace{888\ldots 8}_{n \text{ 8's}} \pmod{10^n} . \]

Find \(\displaystyle \dfrac12 {\sum_{n=1}^{8} x_n} \).

Hint: \(x_1=2\) and \(x_2=42\). The answer is a prime number as well.


Inspiration.

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