If 8947A56B1 is divisible by 9, where B is an odd number. Find the sum of all possible value of A?

Options

A. 26
B. 27
C. 30
D. 36

Answer

To determine the possible values of \( A \) such that the number \( 8947A56B1 \) is divisible by 9, we will follow these steps:

### Step 1: Understand the divisibility rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.

### Step 2: Calculate the sum of the known digits
We will first calculate the sum of the known digits in the number \( 8947A56B1 \):
\[ 8 + 9 + 4 + 7 + 5 + 6 + 1 = 40 \]
So, the sum of the known digits is \( 40 \).

### Step 3: Include \( A \) and \( B \) in the sum
Now, we include \( A \) and \( B \) in the sum:
\[ \text{Total Sum} = 40 + A + B \]

### Step 4: Set up the condition for divisibility by 9
For \( 8947A56B1 \) to be divisible by 9, \( 40 + A + B \) must be divisible by 9.

### Step 5: Determine the possible values of \( B \)
We know \( B \) is an odd number. The possible values for \( B \) are \( 1, 3, 5, 7, 9 \).

### Step 6: Analyze each case for \( B \)
We will analyze each case for \( B \) and find the corresponding values of \( A \).

1. **If \( B = 1 \)**:
\[ 40 + A + 1 = 41 + A \]
We need \( 41 + A \equiv 0 \mod 9 \).
\( 41 \mod 9 = 5 \), so \( A + 5 \equiv 0 \mod 9 \) implies \( A \equiv 4 \mod 9 \). Thus, \( A = 4 \).

2. **If \( B = 3 \)**:
\[ 40 + A + 3 = 43 + A \]
\( 43 \mod 9 = 7 \), so \( A + 7 \equiv 0 \mod 9 \) implies \( A \equiv 2 \mod 9 \). Thus, \( A = 2 \).

3. **If \( B = 5 \)**:
\[ 40 + A + 5 = 45 + A \]
\( 45 \mod 9 = 0 \), so \( A \equiv 0 \mod 9 \). Thus, \( A = 0 \) or \( A = 9 \).

4. **If \( B = 7 \)**:
\[ 40 + A + 7 = 47 + A \]
\( 47 \mod 9 = 2 \), so \( A + 2 \equiv 0 \mod 9 \) implies \( A \equiv 7 \mod 9 \). Thus, \( A = 7 \).

5. **If \( B = 9 \)**:
\[ 40 + A + 9 = 49 + A \]
\( 49 \mod 9 = 4 \), so \( A + 4 \equiv 0 \mod 9 \) implies \( A \equiv 5 \mod 9 \). Thus, \( A = 5 \).

### Step 7: List all possible values of \( A \)
From the above cases, the possible values of \( A \) are:
- \( 4 \) (when \( B = 1 \))
- \( 2 \) (when \( B = 3 \))
- \( 0 \) (when \( B = 5 \))
- \( 9 \) (when \( B = 5 \))
- \( 7 \) (when \( B = 7 \))
- \( 5 \) (when \( B = 9 \))

### Step 8: Calculate the sum of all possible values of \( A \)
Now we sum all the unique values of \( A \):
\[ 0 + 2 + 4 + 5 + 7 + 9 = 27 \]

### Final Answer
The sum of all possible values of \( A \) is \( 27 \).

---