When 335 is added to 5A7, the result is 8B2, 8B2 is divisible by 3. What is the largest possible value of A?

Options

A. 8
B. 5
C. 1
D. 4

Answer

To solve the problem step by step, we need to find the largest possible value of \( A \) such that when \( 335 \) is added to \( 5A7 \), the result \( 8B2 \) is divisible by \( 3 \).

### Step 1: Set up the equation
We start with the equation:
\[ 5A7 + 335 = 8B2 \]

### Step 2: Break down the addition
Let's analyze the addition column by column:
- The units column: \( 7 + 5 = 12 \) (write down \( 2 \) and carry \( 1 \))
- The tens column: \( A + 3 + 1 = A + 4 \) (this is where we will find \( B \))
- The hundreds column: \( 5 + 3 = 8 \)

So we can summarize the addition as:
\[ 5A7 + 335 = 8B2 \quad \Rightarrow \quad 7 + 5 = 2 \text{ (carry 1)}, \quad A + 4 = B, \quad 5 + 3 = 8 \]

### Step 3: Find the relationship between \( A \) and \( B \)
From the addition in the tens column, we have:
\[ B = A + 4 \]
Since \( B \) is a single digit, we must ensure that \( A + 4 < 10 \):
\[ A < 6 \]
Thus, \( A \) can take values \( 0, 1, 2, 3, 4, 5 \).

### Step 4: Check divisibility by 3
Next, we need to ensure that \( 8B2 \) is divisible by \( 3 \). For a number to be divisible by \( 3 \), the sum of its digits must be divisible by \( 3 \):
\[ 8 + B + 2 = 10 + B \]
Substituting \( B = A + 4 \):
\[ 10 + (A + 4) = 14 + A \]
We need \( 14 + A \) to be divisible by \( 3 \).

### Step 5: Check possible values of \( A \)
Now we check the possible values of \( A \):
- If \( A = 0 \): \( 14 + 0 = 14 \) (not divisible by 3)
- If \( A = 1 \): \( 14 + 1 = 15 \) (divisible by 3)
- If \( A = 2 \): \( 14 + 2 = 16 \) (not divisible by 3)
- If \( A = 3 \): \( 14 + 3 = 17 \) (not divisible by 3)
- If \( A = 4 \): \( 14 + 4 = 18 \) (divisible by 3)
- If \( A = 5 \): \( 14 + 5 = 19 \) (not divisible by 3)

### Step 6: Conclusion
The possible values of \( A \) that make \( 8B2 \) divisible by \( 3 \) are \( 1 \) and \( 4 \). The largest possible value of \( A \) is:
\[ \boxed{4} \]