Here is the completed conversion table: | Decimal | Binary | Octal | Hexadecimal | | :------ | :-------- | :---- | :---------- | | 33 | 100001 | 41 | 21 | | 117 | 1110101 | 165 | 75 | | 451 | 111000011 | 703 | 1C3 | | 431 | 110101111 | 657 | 1AF | Here are the step-by-step conversions for each row: Row 1: Given Decimal 33 Decimal to Binary: Step 1: Divide 33 by 2 repeatedly and record the remainders. $$ 33 \div 2 = 16 \text{ R } 1 $$ $$ 16 \div 2 = 8 \text{ R } 0 $$ $$ 8 \div 2 = 4 \text{ R } 0 $$ $$ 4 \div 2 = 2 \text{ R } 0 $$ $$ 2 \div 2 = 1 \text{ R } 0 $$ $$ 1 \div 2 = 0 \text{ R } 1 $$ Step 2: Read the remainders from bottom to top. Binary: $\boxed{\text{100001}}$ Decimal to Octal: Step 1: Divide 33 by 8 repeatedly and record the remainders. $$ 33 \div 8 = 4 \text{ R } 1 $$ $$ 4 \div 8 = 0 \text{ R } 4 $$ Step 2: Read the remainders from bottom to top. Octal: $\boxed{\text{41}}$ Decimal to Hexadecimal: Step 1: Divide 33 by 16 repeatedly and record the remainders. $$ 33 \div 16 = 2 \text{ R } 1 $$ $$ 2 \div 16 = 0 \text{ R } 2 $$ Step 2: Read the remainders from bottom to top. Hexadecimal: $\boxed{\text{21}}$ Row 2: Given Binary 1110101 Binary to Decimal: Step 1: Assign powers of 2 to each binary digit from right to left, starting with $2^0$. $$ 1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 $$ Step 2: Calculate the sum. $$ 64 + 32 + 16 + 0 + 4 + 0 + 1 = 117 $$ Decimal: $\boxed{\text{117}}$ Binary to Octal: Step 1: Group the binary digits into sets of three from right to left. Add leading zeros if necessary. $$ 1 \ 110 \ 101 $$ Step 2: Convert each group of three binary digits to its octal equivalent. $$ 001_2 = 1_8 $$ $$ 110_2 = 6_8 $$ $$ 101_2 = 5_8 $$ Octal: $\boxed{\text{165}}$ Binary to Hexadecimal: Step 1: Group the binary digits into sets of four from right to left. Add leading zeros if necessary. $$ 0111 \ 0101 $$ Step 2: Convert each group of four binary digits to its hexadecimal equivalent. $$ 0111_2 = 7_{16} $$ $$ 0101_2 = 5_{16} $$ Hexadecimal: $\boxed{\text{75}}$ Row 3: Given Octal 703 Octal to Decimal: Step 1: Assign powers of 8 to each octal digit from right to left, starting with $8^0$. $$ 7 \cdot 8^2 + 0 \cdot 8^1 + 3 \cdot 8^0 $$ Step 2: Calculate the sum. $$ 7 \cdot 64 + 0 \cdot 8 + 3 \cdot 1 = 448 + 0 + 3 = 451 $$ Decimal: $\boxed{\text{451}}$ Octal to Binary: Step 1: Convert each octal digit to its 3-bit binary equivalent. $$ 7_8 = 111_2 $$ $$ 0_8 = 000_2 $$ $$ 3_8 = 011_2 $$ Step 2: Combine the binary groups. Binary: $\boxed{\text{111000011}}$ Octal to Hexadecimal: Step 1: Convert Octal to Binary (as above). $$ 703_8 = 111000011_2 $$ Step 2: Group the binary digits into sets of four from right to left. Add leading zeros if necessary. $$ 0001 \ 1100 \ 0011 $$ Step 3: Convert each group of four binary digits to its hexadecimal equivalent. $$ 0001_2 = 1_{16} $$ $$ 1100_2 = C_{16} $$ $$ 0011_2 = 3_{16} $$ Hexadecimal: $\boxed{\text{1C3}}$ Row 4: Given Hexadecimal 1AF Hexadecimal to Decimal: Step 1: Assign powers of 16 to each hexadecimal digit from right to left, starting with $16^0$. (Note: A=10, F=15) $$ 1 \cdot 16^2 + A \cdot 16^1 + F \cdot 16^0 $$ $$ 1 \cdot 16^2 + 10 \cdot 16^1 + 15 \cdot 16^0 $$ Step 2: Calculate the sum. $$ 1 \cdot 256 + 10 \cdot 16 + 15 \cdot 1 = 256 + 160 + 15 = 431 $$ Decimal: $\boxed{\text{431}}$ Hexadecimal to Binary: Step 1: Convert each hexadecimal digit to its 4-bit binary equivalent. $$ 1_{16} = 0001_2 $$ $$ A_{16} = 1010_2 $$ $$ F_{16} = 1111_2 $$ Step 2: Combine the binary groups. Leading zeros can be omitted. Binary: $\boxed{\text{110101111}}$ Hexadecimal to Octal: Step 1: Convert Hexadecimal to Binary (as above). $$ 1AF_{16} = 000110101111_2 $$ Step 2: Group the binary digits into sets of three from right to left. Add leading zeros if necessary. $$ 000 \ 110 \ 101 \ 111 $$ Step 3: Convert each group of three binary digits to its octal equivalent. $$ 000_2 = 0_8 $$ $$ 110_2 = 6_8 $$ $$ 101_2 = 5_8 $$ $$ 111_2 = 7_8 $$ Octal: $\boxed{\text{657}}$