Convert \( \mathrm{A}_{4} 5_{16} \) to octal. \( \mathrm{A} 015_{8} \) \( \mathrm{~B} 115_{8} \) \( 4115_{8} \) \( 5105_{8} \)

To convert the hexadecimal number \( \mathrm{A}_{4} 5_{16} \) to octal, we can follow these steps: 1. **Convert from hexadecimal to decimal.** 2. **Convert from decimal to octal.** ### Step 1: Convert from hexadecimal to decimal The hexadecimal number \( \mathrm{A}_{4} 5_{16} \) can be broken down as follows: - \( \mathrm{A} = 10 \) - \( 4 = 4 \) - \( 5 = 5 \) The positional values in hexadecimal (base 16) are \( 16^2, 16^1, \) and \( 16^0 \) respectively. Therefore, we can calculate the decimal value: \[ \text{Decimal} = A \cdot 16^2 + 4 \cdot 16^1 + 5 \cdot 16^0 \] \[ = 10 \cdot 256 + 4 \cdot 16 + 5 \cdot 1 \] \[ = 2560 + 64 + 5 \] \[ = 2629 \] ### Step 2: Convert from decimal to octal Now we need to convert \( 2629 \) from decimal to octal. We do this by repeatedly dividing by \( 8 \) and keeping track of the remainders. 1. \( 2629 \div 8 = 328 \) remainder \( 5 \) 2. \( 328 \div 8 = 41 \) remainder \( 0 \) 3. \( 41 \div 8 = 5 \) remainder \( 1 \) 4. \( 5 \div 8 = 0 \) remainder \( 5 \) Now we read the remainders from bottom to top to get the octal representation: \[ 2629_{10} = 5105_{8} \] ### Conclusion Thus, the octal representation of \( \mathrm{A}_{4} 5_{16} \) is \( 5105_{8} \). The correct answer is: \[ \boxed{5105_{8}} \]