### Instruction: Find the product of $123_4 \cdot 3_4$.  Express your answer in base 4.### Response: I know that base 4 means each digit represents a power of 4, so for example, $123_4 = 1 \cdot 4^2 + 2 \cdot 4^1 + 3 \cdot 4^0 = 27_{10}$.
Similarly, $3_4 = 3 \cdot 4^0 = 3_{10}$.
To find the product of $123_4 \cdot 3_4$, I could convert both numbers to base 10, multiply them, and then convert the result back to base 4.
So, $123_4 \cdot 3_4 = 27_{10} \cdot 3_{10} = 81_{10}$.
To convert 81 from base 10 to base 4, I need to find the largest power of 4 that is less than or equal to 81, which is $4^3 = 64$.
Then, I need to subtract 64 from 81 and repeat the process with the remaining powers of 4.
So, $81 - 64 = 17$, and the largest power of 4 that is less than or equal to 17 is $4^2 = 16$.
Then, $17 - 16 = 1$, and the largest power of 4 that is less than or equal to 1 is $4^0 = 1$.
This means that $81_{10} = 1 \cdot 4^3 + 0 \cdot 4^2 + 1 \cdot 4^1 + 1 \cdot 4^0 = 1011_4$.
Therefore, the final answer is $1011_4$.