When the base-16 number $66666_{16}$ is written in base 2, how many base-2 digits (bits) does it have?

Amanita, let's solve the problem step by step:

Step 1: Understand the Problem
We are given a number in base-16 (hexadecimal): $66666_{16}$. We need to find out how many digits (bits) this number will have when written in base-2 (binary).

Step 2: Convert the Base-16 Number to Base-10
First, let's convert $66666_{16}$ to base-10 (decimal). This will help us work with the number more easily.
The formula for converting from base-16 to base-10 is:

$66666_{16} = 6 \times 16^4 + 6 \times 16^3 + 6 \times 16^2 + 6 \times 16^1 + 6 \times 16^0$

Step 3: Find the Number of Binary Digits (Bits)
Once we have the decimal value, the number of binary digits (bits) required to represent a number $n$ is given by:

$\text{Number of bits} = \lfloor \log_2 n \rfloor + 1$

where $\lfloor x \rfloor$ means the greatest integer less than or equal to $x$.

Let's execute these steps.

Step 2: Convert $66666_{16}$ to decimal

$66666_{16} = 6 \times 16^4 + 6 \times 16^3 + 6 \times 16^2 + 6 \times 16^1 + 6 \times 16^0$

Let's calculate this value. Amanita, here’s how we solve the problem:

1. First, we convert the base-16 number $66666_{16}$ to base-10. Using the formula for base conversion, we get:

$66666_{16} = 6 \times 16^4 + 6 \times 16^3 + 6 \times 16^2 + 6 \times 16^1 + 6 \times 16^0 = 419430$

2. Next, to find out how many binary digits (bits) are needed to write this number in base-2, we use the formula:

$\text{Number of bits} = \lfloor \log_2{419430} \rfloor + 1$

Calculating this, we find the answer is $13$.

Final Answer:
When the base-16 number $66666_{16}$ is written in base-2, it has $13$ binary digits (bits).