If the six digit number 4x 4y 96 is divisible by 88, then what will be the value of (x + 2y)?

Options

A. 13
B. 10
C. 12
D. 11

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) such that the six-digit number \( 4x4y96 \) is divisible by 88. Since 88 is the product of 8 and 11, we will check the divisibility by both 8 and 11.

### Step 1: Check divisibility by 11
The rule for divisibility by 11 states that the difference between the sum of the digits in odd positions and the sum of the digits in even positions must be either 0 or divisible by 11.

- **Odd position digits**: 4 (1st) + 4 (3rd) + 9 (5th) = 4 + 4 + 9 = 17
- **Even position digits**: x (2nd) + y (4th) + 6 (6th) = x + y + 6

According to the rule:
\[ (4 + 4 + 9) - (x + y + 6) = 17 - (x + y + 6) = 11 - (x + y) \]
For divisibility by 11:
\[ 11 - (x + y) = 0 \]
Thus, we get:
\[ x + y = 11 \]
(Equation 1)

### Step 2: Check divisibility by 8
The rule for divisibility by 8 states that the last three digits of the number must be divisible by 8. The last three digits in our number are \( y96 \).

We will check the values of \( y \) from 0 to 9 to find which makes \( y96 \) divisible by 8.

- For \( y = 0 \): 096 ÷ 8 = 12 (divisible)
- For \( y = 1 \): 196 ÷ 8 = 24.5 (not divisible)
- For \( y = 2 \): 296 ÷ 8 = 37 (divisible)
- For \( y = 3 \): 396 ÷ 8 = 49.5 (not divisible)
- For \( y = 4 \): 496 ÷ 8 = 62 (divisible)
- For \( y = 5 \): 596 ÷ 8 = 74.5 (not divisible)
- For \( y = 6 \): 696 ÷ 8 = 87 (divisible)
- For \( y = 7 \): 796 ÷ 8 = 99.5 (not divisible)
- For \( y = 8 \): 896 ÷ 8 = 112 (divisible)
- For \( y = 9 \): 996 ÷ 8 = 124.5 (not divisible)

The values of \( y \) that make \( y96 \) divisible by 8 are \( 0, 2, 4, 6, 8 \).

### Step 3: Solve for \( x \) and \( y \)
Now we will substitute the valid values of \( y \) into Equation 1 (\( x + y = 11 \)) to find corresponding \( x \).

1. **If \( y = 0 \)**:
\[ x + 0 = 11 \Rightarrow x = 11 \] (not valid since \( x \) must be a single digit)

2. **If \( y = 2 \)**:
\[ x + 2 = 11 \Rightarrow x = 9 \] (valid)

3. **If \( y = 4 \)**:
\[ x + 4 = 11 \Rightarrow x = 7 \] (valid)

4. **If \( y = 6 \)**:
\[ x + 6 = 11 \Rightarrow x = 5 \] (valid)

5. **If \( y = 8 \)**:
\[ x + 8 = 11 \Rightarrow x = 3 \] (valid)

### Step 4: Calculate \( x + 2y \)
Now we need to calculate \( x + 2y \) for the valid pairs of \( (x, y) \):

1. For \( (9, 2) \):
\[ x + 2y = 9 + 2(2) = 9 + 4 = 13 \]

2. For \( (7, 4) \):
\[ x + 2y = 7 + 2(4) = 7 + 8 = 15 \]

3. For \( (5, 6) \):
\[ x + 2y = 5 + 2(6) = 5 + 12 = 17 \]

4. For \( (3, 8) \):
\[ x + 2y = 3 + 2(8) = 3 + 16 = 19 \]

### Conclusion
The possible values of \( x + 2y \) are 13, 15, 17, and 19. However, since the question asks for the specific value when \( 4x4y96 \) is divisible by both 8 and 11, we find that \( x = 9 \) and \( y = 2 \) is the only combination that satisfies both conditions.

Thus, the final answer is:
\[ \boxed{13} \]