Question 5
List $A$ contains the numbers $10^k$ in base $10$, for integers $k\ge1$. Lists $B,C$ contain the same numbers written in bases $2,5$, respectively. The first entries are: $10\leftrightarrow1010_2\leftrightarrow20_5$; $100\leftrightarrow1100100_2\leftrightarrow400_5$; $1000\leftrightarrow1111101000_2\leftrightarrow13000_5$.
Prove that for every integer $n\gt 1$, exactly one entry in exactly one of the lists $B,C$ has exactly $n$ digits.
Prove that for every integer $n\gt 1$, exactly one entry in exactly one of the lists $B,C$ has exactly $n$ digits.