The sum of the binary numbers `(11011) (10110110)_(2) and (10011x0y)_(2)` is the binary number `(101101101)_(2)` What are the value of x and y?

Options

A. a) x = 1. y = 1
B. b) x= 1. y = 0
C. c) x = 0, y = 1
D. d) x = 0, y = 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. **Convert the Binary Numbers to Decimal**:
- Convert \( (11011)_2 \):
\[ 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 16 + 8 + 0 + 2 + 1 = 27 \]
- Convert \( (10110110)_2 \):
\[ 1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 = 128 + 0 + 32 + 16 + 0 + 4 + 2 + 0 = 182 \]
- Convert \( (101101101)_2 \):
\[ 1 \cdot 2^8 + 0 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 256 + 0 + 64 + 32 + 0 + 8 + 0 + 2 + 1 = 363 \]

2. **Set Up the Equation**:
- We know that:
\[ (11011)_2 + (10110110)_2 + (10011x0y)_2 = (101101101)_2 \]
- In decimal, this can be expressed as:
\[ 27 + 182 + (10011x0y)_2 = 363 \]
- Therefore, we can find \( (10011x0y)_2 \):
\[ (10011x0y)_2 = 363 - 27 - 182 = 154 \]

3. **Convert 154 to Binary**:
- To convert \( 154 \) to binary:
- \( 154 \div 2 = 77 \) remainder \( 0 \)
- \( 77 \div 2 = 38 \) remainder \( 1 \)
- \( 38 \div 2 = 19 \) remainder \( 0 \)
- \( 19 \div 2 = 9 \) remainder \( 1 \)
- \( 9 \div 2 = 4 \) remainder \( 1 \)
- \( 4 \div 2 = 2 \) remainder \( 0 \)
- \( 2 \div 2 = 1 \) remainder \( 0 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)
- Reading the remainders from bottom to top gives us \( (10011010)_2 \).

4. **Identify \( x \) and \( y \)**:
- From \( (10011x0y)_2 \) we have:
\[ 10011x0y = 10011010 \]
- Comparing both, we find:
- \( x = 1 \)
- \( y = 0 \)

### Final Answer:
The values of \( x \) and \( y \) are:
\[ x = 1, \quad y = 0 \]