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. 2
C. 1
D. 4

Answer

To solve the problem step by step, we need to find the largest possible value of A in the equation where 335 is added to 5A7, resulting in 8B2, and where 8B2 is divisible by 3.

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

### Step 2: Perform the addition
Let's break down the addition:
- The units place: \( 7 + 5 = 12 \) (write down 2 and carry over 1)
- The tens place: \( A + 3 + 1 = A + 4 \)
- The hundreds place: \( 5 + 3 = 8 \)

So, we can express this as:
\[ 5A7 + 335 = 8(A + 4)2 \]

### Step 3: Analyze the equation
From the addition, we can see that:
- The hundreds digit is confirmed as 8.
- The units digit is confirmed as 2.
- The tens digit gives us \( B = A + 4 \).

### Step 4: Check divisibility by 3
For \( 8B2 \) to be divisible by 3, the sum of its digits must be divisible by 3. Therefore:
\[ 8 + B + 2 = 10 + B \]

### Step 5: Substitute \( B \)
Substituting \( B \) with \( A + 4 \):
\[ 10 + (A + 4) = A + 14 \]

### Step 6: Check divisibility condition
Now, we need \( A + 14 \) to be divisible by 3. We can check the values of \( A \) from 0 to 9 to find the largest possible value of \( A \).

### Step 7: Check values of A
1. If \( A = 0 \): \( 0 + 14 = 14 \) (not divisible by 3)
2. If \( A = 1 \): \( 1 + 14 = 15 \) (divisible by 3)
3. If \( A = 2 \): \( 2 + 14 = 16 \) (not divisible by 3)
4. If \( A = 3 \): \( 3 + 14 = 17 \) (not divisible by 3)
5. If \( A = 4 \): \( 4 + 14 = 18 \) (divisible by 3)
6. If \( A = 5 \): \( 5 + 14 = 19 \) (not divisible by 3)
7. If \( A = 6 \): \( 6 + 14 = 20 \) (not divisible by 3)
8. If \( A = 7 \): \( 7 + 14 = 21 \) (divisible by 3)
9. If \( A = 8 \): \( 8 + 14 = 22 \) (not divisible by 3)
10. If \( A = 9 \): \( 9 + 14 = 23 \) (not divisible by 3)

### Step 8: Identify the largest A
The values of \( A \) that satisfy the divisibility condition are 1, 4, and 7. The largest possible value of \( A \) is:
\[ \text{Largest A} = 7 \]

### Final Answer
The largest possible value of A is **7**.