### 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 1023 in base 4 is equivalent to $1\cdot 4^3 + 0\cdot 4^2 + 2\cdot 4^1 + 3\cdot 4^0 = 64 + 0 + 8 + 3 = 75$ in base 10.
Similarly, 11 in base 4 is equivalent to $1\cdot 4^1 + 1\cdot 4^0 = 4 + 1 = 5$ in base 10.
So the quotient $1023_4 \div 11_4$ is the same as $75 \div 5$ in base 10, which is 15 in base 10.
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 divide 15 by 4 and get a quotient of 3 and a remainder of 3.
So the first digit of the base 4 representation is 3, and the remainder is what I need to convert next.
To convert 3 to base 4, I need to find the largest power of 4 that is less than or equal to 3, which is $4^0 = 1$.
Then I divide 3 by 1 and get a quotient of 3 and a remainder of 0.
So the second digit of the base 4 representation is 3, and the remainder is zero, which means I'm done.