In the binary equation `(1p101)_(2) + (10q1)_(2) = (100r 00)_(2)` where p,q and r are binary digits, what are the possible values of p,q and r respectively?

Options

A. 0,1,0
B. 1,1,0
C. 0,0,1
D. 1,0,1

Answer

To solve the binary equation \( (1p101)_{2} + (10q1)_{2} = (100r00)_{2} \), we will convert each binary number into decimal form and then solve for the binary digits \( p, q, \) and \( r \).

### Step 1: Convert \( (1p101)_{2} \) to Decimal
The binary number \( (1p101)_{2} \) can be expressed in decimal as follows:
\[ 1 \times 2^4 + p \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 \]
Calculating each term:
\[ = 16 + 8p + 4 + 0 + 1 = 21 + 8p \]

### Step 2: Convert \( (10q1)_{2} \) to Decimal
Next, we convert \( (10q1)_{2} \):
\[ 1 \times 2^3 + 0 \times 2^2 + q \times 2^1 + 1 \times 2^0 \]
Calculating each term:
\[ = 8 + 0 + 2q + 1 = 9 + 2q \]

### Step 3: Convert \( (100r00)_{2} \) to Decimal
Now we convert \( (100r00)_{2} \):
\[ 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + r \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \]
Calculating each term:
\[ = 32 + 0 + 0 + 4r + 0 + 0 = 32 + 4r \]

### Step 4: Set Up the Equation
Now we can set up the equation based on the conversions:
\[ (21 + 8p) + (9 + 2q) = (32 + 4r) \]
Combining like terms gives:
\[ 30 + 8p + 2q = 32 + 4r \]

### Step 5: Rearranging the Equation
Rearranging the equation to isolate terms involving \( p, q, \) and \( r \):
\[ 8p + 2q = 4r + 32 - 30 \]
This simplifies to:
\[ 8p + 2q = 4r + 2 \]

### Step 6: Simplifying Further
Dividing the entire equation by 2:
\[ 4p + q = 2r + 1 \]

### Step 7: Finding Possible Values for \( p, q, r \)
Since \( p, q, r \) are binary digits, they can only be \( 0 \) or \( 1 \). We will try different combinations of \( p, q, \) and \( r \) to satisfy the equation \( 4p + q = 2r + 1 \).

1. **If \( p = 0 \)**:
\[ 4(0) + q = 2r + 1 \implies q = 2r + 1 \]
- \( r = 0 \) gives \( q = 1 \) (valid)
- \( r = 1 \) gives \( q = 3 \) (invalid)

2. **If \( p = 1 \)**:
\[ 4(1) + q = 2r + 1 \implies 4 + q = 2r + 1 \implies q = 2r - 3 \]
- \( r = 0 \) gives \( q = -3 \) (invalid)
- \( r = 1 \) gives \( q = -1 \) (invalid)

### Conclusion
The only valid solution is:
- \( p = 0 \)
- \( q = 1 \)
- \( r = 0 \)

Thus, the possible values of \( p, q, r \) are \( 0, 1, 0 \) respectively.