GATE CS 2003 Q9 — Number Representations (Fixed and Floating Point)

To determine which number is divisible by the given divisor, we first convert the numbers from their 8-bit 2's complement representation to decimal.

1. Find the Decimal Value of the Divisor

The given divisor is $11111011_2$. Since the most significant bit (MSB) is $1$, it represents a negative number.

To find its magnitude, we take its 2's complement:

• 1's complement of $11111011$ is $00000100$
• Add $1$: $00000100 + 1 = 00000101_2 = 5_{10}$

Thus, the divisor is $-5$.

2. Convert the Options to Decimal

• Option A: $11100111_2$
  Taking 2's complement to find magnitude:
  $11100111 \xrightarrow{\text{1's comp}} 00011000 \xrightarrow{+1} 00011001_2 = 25_{10}$
  So, the value is $-25$.
  Since $-25$ is divisible by $-5$ ($-25 / -5 = 5$), this option is correct.


• Option B: $11100100_2$
  Taking 2's complement to find magnitude:
  $11100100 \xrightarrow{\text{1's comp}} 00011011 \xrightarrow{+1} 00011100_2 = 28_{10}$
  So, the value is $-28$ (not divisible by $-5$).


• Option C: $11010111_2$
  Taking 2's complement to find magnitude:
  $11010111 \xrightarrow{\text{1's comp}} 00101000 \xrightarrow{+1} 00101001_2 = 41_{10}$
  So, the value is $-41$ (not divisible by $-5$).


• Option D: $11011011_2$
  Taking 2's complement to find magnitude:
  $11011011 \xrightarrow{\text{1's comp}} 00100100 \xrightarrow{+1} 00100101_2 = 37_{10}$
  So, the value is $-37$ (not divisible by $-5$).

Therefore, only the number in Option A is divisible by the divisor.