If the sum of the binary numbers `(11011)_(2) (10110110)_(2) and (10011 x0y)_(2)` is the binary number `(101101101)_(2)` then the values of X and y respectively, are:

Options

A. 1 and 1
B. 1 and 0
C. 0 and 1
D. 0 and 0

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) in the binary number \( (10011x0y)_2 \) such that the sum of the binary numbers \( (11011)_2 \), \( (10110110)_2 \), and \( (10011x0y)_2 \) equals \( (101101101)_2 \).

### Step-by-Step Solution:

1. **Write down the equation:**
We know that:
\[ (11011)_2 + (10110110)_2 + (10011x0y)_2 = (101101101)_2 \]

2. **Convert the binary numbers to decimal for easier addition:**
- \( (11011)_2 = 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 16 + 8 + 0 + 2 + 1 = 27 \)
- \( (10110110)_2 = 1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 128 + 0 + 32 + 16 + 0 + 4 + 2 + 0 = 182 \)
- \( (101101101)_2 = 1 \cdot 2^8 + 0 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1 = 365 \)

3. **Set up the equation in decimal:**
\[ 27 + 182 + (10011x0y)_2 = 365 \]

4. **Calculate the sum of the known values:**
\[ 27 + 182 = 209 \]

5. **Find the decimal value of \( (10011x0y)_2 \):**
\[ (10011x0y)_2 = 365 - 209 = 156 \]

6. **Convert 156 back to binary to find \( x \) and \( y \):**
- Convert 156 to binary:
- 156 divided by 2 gives a quotient of 78 and a remainder of 0.
- 78 divided by 2 gives a quotient of 39 and a remainder of 0.
- 39 divided by 2 gives a quotient of 19 and a remainder of 1.
- 19 divided by 2 gives a quotient of 9 and a remainder of 1.
- 9 divided by 2 gives a quotient of 4 and a remainder of 1.
- 4 divided by 2 gives a quotient of 2 and a remainder of 0.
- 2 divided by 2 gives a quotient of 1 and a remainder of 0.
- 1 divided by 2 gives a quotient of 0 and a remainder of 1.
- Reading the remainders from bottom to top gives us \( 10011100_2 \).

7. **Identify \( x \) and \( y \):**
From \( (10011x0y)_2 = (10011100)_2 \), we can see that:
- \( x = 1 \)
- \( y = 0 \)

### Final Answer:
The values of \( x \) and \( y \) respectively are \( 1 \) and \( 0 \).