### Instruction: Express the quotient $1023_4 \div 11_4$ in base 4.### Response: I know that base 4 means that each digit represents a power of 4, so for example, $1023_4$ means $1 \times 4^3 + 0 \times 4^2 + 2 \times 4^1 + 3 \times 4^0 = 64 + 0 + 8 + 3 = 75$ in base 10.
Similarly, $11_4$ means $1 \times 4^1 + 1 \times 4^0 = 5$ in base 10.
So the problem is equivalent to finding the quotient of 75 and 5 in base 10, which is 15.
To convert 15 back to base 4, I need to find the largest power of 4 that is less than or equal to 15, which is $4^1 = 4$.
Then I need to see how many times 4 goes into 15, which is 3 times with a remainder of 3.
So the first digit of the quotient in base 4 is 3, and the remainder is 3.
To find the next digit, I need to repeat the process with the remainder, but using the next lower power of 4, which is $4^0 = 1$.
Then I need to see how many times 1 goes into 3, which is 3 times with no remainder.
So the second digit of the quotient in base 4 is 3, and there is no more remainder.
Therefore, the quotient in base 4 is 33.