4. Convert each of the following numbers to their respective number systems equivalent: \( \begin{array}{ll}\text { a) } \mathrm{A}_{2} \mathrm{DFE}_{16} \text { to binary. } \\ \text { b) } 11110000_{2} \text { to octal. } & \text { ( } 2 \text { Marks) } \\ \text { c) } 65_{10} \text { to hexadecimal. } & \text { (2 Marks) } \\ \text { d) } 673 \mathrm{~s} \text { to decimal. } & \text { (2 Marks) }\end{array} \)

Let's convert each of the given numbers to their respective number systems. ### a) Convert \( \mathrm{DFE}_{16} \) to binary. To convert from hexadecimal (base 16) to binary (base 2), we convert each hexadecimal digit to its 4-bit binary equivalent: - \( D_{16} = 13_{10} = 1101_{2} \) - \( F_{16} = 15_{10} = 1111_{2} \) - \( E_{16} = 14_{10} = 1110_{2} \) Now, we combine these binary values: \[ \mathrm{DFE}_{16} = 1101 \, 1111 \, 1110_{2} \] So, \( \mathrm{A}_{2} \mathrm{DFE}_{16} \) in binary is: \[ \mathrm{DFE}_{16} = 110111111110_{2} \] ### b) Convert \( 11110000_{2} \) to octal. To convert from binary (base 2) to octal (base 8), we group the binary digits into sets of three, starting from the right: \[ 11110000_{2} = 11 \, 110 \, 000 \] We can add leading zeros to make a complete group of three: \[ 001 \, 111 \, 100 \, 000 \] Now, we convert each group to its octal equivalent: - \( 000_{2} = 0_{8} \) - \( 100_{2} = 4_{8} \) - \( 111_{2} = 7_{8} \) - \( 001_{2} = 1_{8} \) Combining these gives us: \[ 11110000_{2} = 1740_{8} \] ### c) Convert \( 65_{10} \) to hexadecimal. To convert from decimal (base 10) to hexadecimal (base 16), we divide the number by 16 and keep track of the remainders: 1. \( 65 \div 16 = 4 \) remainder \( 1 \) 2. \( 4 \div 16 = 0 \) remainder \( 4 \) Reading the remainders from bottom to top, we get: \[ 65_{10} = 41_{16} \] ### d) Convert \( 673_{s} \) to decimal. To convert from a base \( s \) to decimal, we need to know the base \( s \). Assuming \( s \) is base 10 (as it is not specified), we can directly state: \[ 673_{10} = 673_{10} \] If \( s \) is a different base, please specify the base for accurate conversion. ### Summary of Results: - a) \( \mathrm{DFE}_{16} = 110111111110_{2} \) - b) \( 11110000_{2} = 1740_{8} \) - c) \( 65_{10} = 41_{16} \) - d) \( 673_{s} = 673_{10} \) (assuming \( s = 10 \))