The sum of all possible values of a, for which the 4-digit number 556a is divişible by 3, is

Options

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

Answer

To find the sum of all possible values of \( a \) for which the 4-digit number \( 556a \) 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 digits
The digits of the number \( 556a \) are \( 5, 5, 6, \) and \( a \).
So, the sum of the digits is:
\[ 5 + 5 + 6 + a = 16 + a \]

### Step 3: Set up the condition for divisibility
We need \( 16 + a \) to be divisible by 3.

### Step 4: Find the possible values of \( a \)
To find the values of \( a \), we can check what values of \( a \) make \( 16 + a \) divisible by 3.

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

For \( 16 + a \) to be divisible by 3, we need:
\[ 1 + a \equiv 0 \mod 3 \implies a \equiv -1 \equiv 2 \mod 3 \]

### Step 5: Determine the values of \( a \)
The values of \( a \) that satisfy \( a \equiv 2 \mod 3 \) and are single digits (0 to 9) are:
- \( a = 2 \)
- \( a = 5 \)
- \( a = 8 \)

### Step 6: Calculate the sum of all possible values of \( a \)
Now, we sum these values:
\[ 2 + 5 + 8 = 15 \]

### Conclusion
The sum of all possible values of \( a \) for which the number \( 556a \) is divisible by 3 is \( 15 \).