\hspace{-16}$ Prove that there exists a power of the number $\bf{2}$ such that the last\\\\ $\bf{1000}$ digits in its decimal representation are all $\bf{1}$ and $\bf{2}$
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\hspace{-16}$ Prove that there exists a power of the number $\bf{2}$ such that the last\\\\ $\bf{1000}$ digits in its decimal representation are all $\bf{1}$ and $\bf{2}$