How many power of 2?

Number Theory Level 2

\[\large{0.\quad 000001\quad 000002\quad 000004\quad 000008 \ldots}\]

The Above shows the first few digits of the decimal representation of the fraction \(\Large{\frac { 1 }{ 999,998 } }\). If we split the digits into partitions of 6, we can see that the numbers are all powers of 2. Inclusive of \(2^0=1\), how many powers of 2 can we find before the pattern breaks off?

Bonus:- Generalize for \({\dfrac{1}{10^k-2}}\) and \( \dfrac{1}{10^{k}-n}\), where \(k\) and \(n\) are positive integers.

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