# How many power of 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.

×