To solve the problem of finding the remainder \( r \) when \( 321_4 \) is divided by \( 23_4 \), we will follow a step-by-step approach. ### Step 1: Convert the numbers from base 4 to base 10 First, we need to convert both \( 321_4 \) and \( 23_4 \) from base 4 to base 10. **Converting \( 321_4 \):** - The digits from left to right are \( 3, 2, 1 \). - The place values from left to right are \( 4^2, 4^1, 4^0 \). Calculating the base 10 value: \[ 321_4 = 3 \times 4^2 + 2 \times 4^1 + 1 \times 4^0 \] \[ = 3 \times 16 + 2 \times 4 + 1 \times 1 \] \[ = 48 + 8 + 1 = 57 \] **Converting \( 23_4 \):** - The digits from left to right are \( 2, 3 \). - The place values from left to right are \( 4^1, 4^0 \). Calculating the base 10 value: \[ 23_4 = 2 \times 4^1 + 3 \times 4^0 \] \[ = 2 \times 4 + 3 \times 1 \] \[ = 8 + 3 = 11 \] ### Step 2: Perform the division in base 10 Now we need to divide \( 57 \) by \( 11 \) and find the remainder \( r \). Performing the division: \[ 57 \div 11 = 5 \quad \text{(since } 11 \times 5 = 55\text{)} \] Calculating the remainder: \[ r = 57 - 55 = 2 \] ### Step 3: Conclusion Thus, the remainder \( r \) when \( 321_4 \) is divided by \( 23_4 \) is \( 2 \). ### Step 4: Analyze the options Now, let's look at the options provided: - A. zero - B. 1 - C. 2 - D. 3 The correct answer is **C. 2**. ### Why the other options are incorrect: - **Option A (zero)**: This would imply that \( 321_4 \) is exactly divisible by \( 23_4 \), which is not the case since \( 57 \) is not a multiple of \( 11 \). - **Option B (1)**: This would suggest that when \( 57 \) is divided by \( 11 \), the remainder is \( 1 \). However, as calculated, the remainder is \( 2 \). - **Option D (3)**: This would imply that \( 57 \) is \( 3 \) more than a multiple of \( 11 \). Since \( 55 \) is the closest multiple of \( 11 \) to \( 57 \), the remainder cannot be \( 3 \). ### Summary of Key Points: - Convert numbers from base 4 to base 10 for easier calculations. - Perform the division in base 10 to find the quotient and remainder. - The correct remainder when \( 321_4 \) is divided by \( 23_4 \) is \( 2 \). - Always check each option against the calculations to confirm the correct answer. This thorough approach ensures a clear understanding of the problem and the solution process.