Question 161
Prove, that for every integer $k \ge 1$ there is a positive integer $n$ with following property: in decimal representation of $2^n$ one can find a block of exactly $k$ consecutive zeros, i.e.
$$
2^n = \dots a \overline{00\dots0} b\dots, \quad \text{k zeros}
$$
where $a$, $b$ are nonzero digits.
$$
2^n = \dots a \overline{00\dots0} b\dots, \quad \text{k zeros}
$$
where $a$, $b$ are nonzero digits.