1. **Problem:** Express the decimal number $X=380$ in bases 2, 3, 7, 8, and 16. 2. **Formula and rules:** To convert a decimal number to another base $b$, repeatedly divide the number by $b$ and record the remainders. The base-$b$ number is the remainders read in reverse order. 3. **Convert 380 to base 2:** - $380 \div 2 = 190$ remainder $0$ - $190 \div 2 = 95$ remainder $0$ - $95 \div 2 = 47$ remainder $1$ - $47 \div 2 = 23$ remainder $1$ - $23 \div 2 = 11$ remainder $1$ - $11 \div 2 = 5$ remainder $1$ - $5 \div 2 = 2$ remainder $1$ - $2 \div 2 = 1$ remainder $0$ - $1 \div 2 = 0$ remainder $1$ Reading remainders backward: $101111100_2$ 4. **Convert 380 to base 3:** - $380 \div 3 = 126$ remainder $2$ - $126 \div 3 = 42$ remainder $0$ - $42 \div 3 = 14$ remainder $0$ - $14 \div 3 = 4$ remainder $2$ - $4 \div 3 = 1$ remainder $1$ - $1 \div 3 = 0$ remainder $1$ Reading backward: $110202_3$ 5. **Convert 380 to base 7:** - $380 \div 7 = 54$ remainder $2$ - $54 \div 7 = 7$ remainder $5$ - $7 \div 7 = 1$ remainder $0$ - $1 \div 7 = 0$ remainder $1$ Reading backward: $1052_7$ 6. **Convert 380 to base 8:** - $380 \div 8 = 47$ remainder $4$ - $47 \div 8 = 5$ remainder $7$ - $5 \div 8 = 0$ remainder $5$ Reading backward: $574_8$ 7. **Convert 380 to base 16:** - $380 \div 16 = 23$ remainder $12$ (C in hex) - $23 \div 16 = 1$ remainder $7$ - $1 \div 16 = 0$ remainder $1$ Reading backward: $17C_{16}$ **Final answers:** - Base 2: $101111100_2$ - Base 3: $110202_3$ - Base 7: $1052_7$ - Base 8: $574_8$ - Base 16: $17C_{16}$