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 will follow the instructions provided in the video transcript while ensuring clarity in each step.

### Step 1: Set Up the Equation
We know that when 335 is added to 5A7, the result is 8B2. We can express this mathematically as:
\[ 5A7 + 335 = 8B2 \]

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

Thus, we have:
\[ 8B2 \text{ where } B = A + 4 \]

### Step 3: Determine the Values of B
Since \( 8B2 \) must be divisible by 3, we need to find the sum of its digits:
\[ 8 + B + 2 = 10 + B \]
This sum must be divisible by 3.

### Step 4: Find Possible Values for B
Now we will check for values of B (which is a digit from 0 to 9) that make \( 10 + B \) divisible by 3:
- If \( B = 0 \): \( 10 + 0 = 10 \) (not divisible by 3)
- If \( B = 1 \): \( 10 + 1 = 11 \) (not divisible by 3)
- If \( B = 2 \): \( 10 + 2 = 12 \) (divisible by 3)
- If \( B = 3 \): \( 10 + 3 = 13 \) (not divisible by 3)
- If \( B = 4 \): \( 10 + 4 = 14 \) (not divisible by 3)
- If \( B = 5 \): \( 10 + 5 = 15 \) (divisible by 3)
- If \( B = 6 \): \( 10 + 6 = 16 \) (not divisible by 3)
- If \( B = 7 \): \( 10 + 7 = 17 \) (not divisible by 3)
- If \( B = 8 \): \( 10 + 8 = 18 \) (divisible by 3)
- If \( B = 9 \): \( 10 + 9 = 19 \) (not divisible by 3)

The possible values for B are 2, 5, and 8.

### Step 5: Calculate Corresponding Values for A
Now we will find the corresponding values of A for each valid B:
1. **If \( B = 2 \)**:
\[ A + 4 = 2 \]
\[ A = 2 - 4 = -2 \] (not valid)

2. **If \( B = 5 \)**:
\[ A + 4 = 5 \]
\[ A = 5 - 4 = 1 \] (valid)

3. **If \( B = 8 \)**:
\[ A + 4 = 8 \]
\[ A = 8 - 4 = 4 \] (valid)

### Step 6: Determine the Largest Possible Value of A
The valid values of A from our calculations are 1 and 4. The largest possible value of A is:
\[ \text{Largest possible value of A} = 4 \]

### Final Answer
The largest possible value of A is **4**.

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