Question

The complement of \( 27_{10} \) is :

• A. \( 37_{10} \)
• B. \( 62_{10} \)
• C. \( 72_{10} \)
• D. \( 73_{10} \)

Hint:

Shortcut for finding decimal complements: \[ \text{10's complement} = 10^n - N \] For \(27\): \[ 100 - 27 = 73 \] Also remember: \[ \text{10's complement} = (\text{9's complement}) + 1 \]

Answer 1

The Correct Option is D

Concept: In decimal number systems, the complement generally refers to the 9's complement or the 10's complement. For a two-digit decimal number:
• 9's complement is obtained by subtracting each digit from 9.
• 10's complement is obtained by adding 1 to the 9's complement. Mathematically, \[ \text{10's complement of an } n\text{-digit number } N = 10^n - N \]

Step 1: Identify the given number.
The given decimal number is: \[ 27_{10} \] It is a two-digit number, so: \[ n = 2 \]

Step 2: Find the 9's complement.
Subtract each digit from 9: \[ 9 - 2 = 7 \] \[ 9 - 7 = 2 \] Thus, \[ \text{9's complement of } 27 = 72 \]

Step 3: Find the 10's complement.
Add 1 to the 9's complement: \[ 72 + 1 = 73 \] Therefore, \[ \text{10's complement of } 27_{10} = 73_{10} \]

Step 4: Match with the options.
The correct option is: \[ \boxed{73_{10}} \] Hence, the correct answer is: \[ \boxed{(4)\ 73_{10}} \]