The sum of all possible values of a, for which the 4-digit number 547a divisible by 3, is:

Options

A. 13
B. 15
C. 7
D. 10

Answer

To determine the sum of all possible values of \( a \) for which the 4-digit number \( 547a \) is divisible by 3, we can follow these steps:

### Step 1: Understand the divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.

### Step 2: Calculate the sum of the known digits
The digits of the number \( 547a \) are 5, 4, 7, and \( a \). First, we calculate the sum of the known digits:

\[ 5 + 4 + 7 = 16 \]

### Step 3: Include \( a \) in the sum
Now, we include \( a \) in the sum:

\[ 16 + a \]

### Step 4: Set up the divisibility condition
For \( 547a \) to be divisible by 3, the sum \( 16 + a \) must be divisible by 3. We can express this condition mathematically:

\[ 16 + a \equiv 0 \ (\text{mod} \ 3) \]

### Step 5: Find \( 16 \mod 3 \)
Next, we calculate \( 16 \mod 3 \):

\[ 16 \div 3 = 5 \quad \text{(remainder 1)} \]
Thus,
\[ 16 \equiv 1 \ (\text{mod} \ 3) \]

### Step 6: Substitute into the divisibility condition
Now we substitute this back into our condition:

\[ 1 + a \equiv 0 \ (\text{mod} \ 3) \]

### Step 7: Solve for \( a \)
This simplifies to:

\[ a \equiv -1 \equiv 2 \ (\text{mod} \ 3) \]

### Step 8: Determine possible values of \( a \)
The possible values of \( a \) that satisfy \( a \equiv 2 \ (\text{mod} \ 3) \) within the range of a single digit (0 to 9) are:

- \( a = 2 \)
- \( a = 5 \)
- \( a = 8 \)

### Step 9: Calculate the sum of possible values of \( a \)
Now, we sum these possible values:

\[ 2 + 5 + 8 = 15 \]

### Final Answer
The sum of all possible values of \( a \) for which the 4-digit number \( 547a \) is divisible by 3 is:

\[ \boxed{15} \]

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