### Instruction: Convert the base-$64$ number $100_{64}$ to base $62$.### Response: To convert from base $64$ to base $62$, I need to first convert to base $10$ and then to base $62$.
To convert from base $64$ to base $10$, I can use the place value method: $100_{64} = 1 \times 64^2 + 0 \times 64^1 + 0 \times 64^0 = 4096_{10}$.
To convert from base $10$ to base $62$, I can use the repeated division method: I divide $4096$ by $62$ and write down the remainder, then repeat with the quotient until I get a quotient of zero.
So, $4096 \div 62 = 66$ with a remainder of $4$, which means the rightmost digit of the base $62$ number is $4$.
Then, $66 \div 62 = 1$ with a remainder of $4$, which means the next digit is also $4$.
Finally, $1 \div 62 = 0$ with a remainder of $1$, which means the leftmost digit is $1$.
Therefore, the base $62$ number is $144_{62}$.