Question

The decimal number system uses the characters \(0, 1, 2, ..., 8, 9,\) and the octal number system uses the characters \(0, 1, 2, ..., 6, 7.\) For example, the decimal number \(12 (= 1 \times 10^1 + 2 \times 10^0)\) is expressed as \(14 (= 1 \times 8^1 + 4 \times 8^0)\) in the octal number system. The decimal number 108 in the octal number system is:

• A. \(168\)
• B. \(108\)
• C. \(150\)
• D. \(154\)

Hint:

In the octal system, base 8 digits range from 0 to 7. Dividing by 8 repeatedly simplifies conversion.

Answer 1

The Correct Option is D

Step 1: Decimal to octal conversion using division by 8. Convert Decimal to Binary: Divide the number by 2. Get the integer quotient for the next iteration. Get the remainder for the binary digit. Repeat the above steps until the quotient is equal to 0. Given: Decimal Number = 108 \[ 108_{10} \implies \text{Binary Conversion Steps:} \] \[ \begin{array}{|c|c|} \hline \text{Quotient} & \text{Remainder (Binary Digit)}
\hline 108 \div 2 = 54 & 0
54 \div 2 = 27 & 0
27 \div 2 = 13 & 1
13 \div 2 = 6 & 1
6 \div 2 = 3 & 0
3 \div 2 = 1 & 1
1 \div 2 = 0 & 1
\hline \end{array} \] Thus, \( 108_{10} = (1101100)_2 \). Convert Binary to Octal: Group the binary digits into sets of three from right to left: \[ (001\ 101\ 100)_2 \] Convert each group into octal: \[ (001)_2 = 1, \quad (101)_2 = 5, \quad (100)_2 = 4 \] Thus: \[ (1101100)_2 = (154)_8 \] Final Answer: \( 108_{10} = 154_8 \). Reading the remainders from bottom to top gives: \[ (108)_{10} = (154)_8 \] Step 2: Verification. \[ 1 \times 8^2 + 5 \times 8^1 + 4 \times 8^0 = 64 + 40 + 4 = 108. \] Therefore, (4) \(154\) is correct.